Showing posts with label 8 year old. Show all posts

Monday, March 6, 2017

Cryptarithmetic puzzles follow-up

I was asked to write a bit about strategies and answers for the puzzles we gave two weeks ago.

BIG + PIG = YUM
Because the digits in YUM are all distinct from BIG and PIG and there are only 7 letters in this puzzle, we should expect there to be many solutions.

The easiest way to get a feel for the puzzle is to start trying values and see what develops. This was part of the idea of using this puzzle as the opening challenge.

As we play with examples, the kids should notice these things that constrain our possible solutions:

B, G, I, M, P, U, Y must all be distinct
We are adding two three digit numbers and the sum is a three digit number
B, P, and Y are all leading digits
The largest sum possible with two numbers 0 to 9 is 18.

Some conclusions:

(a) G is not 0. If it was, then M would also be 0.

(b) B, P, and Y are all not 0. They are leading digits, the rules of our puzzles say they can't be zero.

(c) G + G is at most 18. It may contribute at most one ten to the calculation of U. That will only happen if G is 5 or larger.

(d) I + I is at most 18. Along with a potential ten from G+G, that means we have at most 19 coming from the tens. That will only happen if I is 5 or larger.

(e) B+G is at most 9. If there is an extra hundred coming from the tens digits, B+ G is at most 8.

(f) If I is 9, G must be less than 5. Can you see why?

(g) If G is less than 5, I cannot be 0

After these observations, I'd suggest picking values of G, then seeing what values of I are allowed, then checking what remains for B and P. Because we aren't allowed to have duplicates, we quickly see that our choices are constrained.

For example, if G is 1 or 2, then I is at least 3 and we get the following possible solutions (B and P can be interchanged):

431 + 531 = 962

341 + 641 = 982

351 + 451 = 802

371 + 571 = 942

381 + 581 = 962

132 + 732 = 864

132 + 832 = 964

152 + 652 = 804

152 + 752 = 904

182 + 582 = 764

192 + 392 = 584

192 + 592 = 784

There are some more advanced ideas that could come out of trying to count or list all of the solutions, so I'd encourage people to explore. Even this simple puzzle can be a lot of fun!

CAT + HAT = BAD

The A in BAD is the key part of this puzzle. We can get two cases:

(a) A is 0 and T is 1, 2, 3 or 4

(b) A is 9 and T is 5, 6, 7 or 8.

Again, while there are a lot of solutions (and counting them would be a fun challenge) they are easiest to build up by choosing A (either 0 or 9), then T, then seeing what flexibility is left for C and H. Here are some examples:

301 + 401 = 702

301 + 501 = 802

301 + 601 = 902

302 + 502 = 804

302 + 602 = 904

103 + 403 = 506

395 + 495 = 890

SAD + MAD + DAD = SORRY

This was a puzzle without a solution. In this case, it isn't too hard to see that SORRY has too many digits. The best explanation was given by one student:

The largest three digit number is 999.
If we add three of them, we will at most get 2997.
SORRY has to be bigger than 10,000.
This isn't possible

CURRY + RICE = LUNCH

Unfortunately, this also doesn't have a solution, but the reasoning is more subtle than the previous puzzle.

Here, we can reason as follows:

R cannot be 0 because it is the leading digit in RICE
Because the tens digit of RICE and LUNCH are both C, R must be 9 and we must have Y + E > 10.
This also means R + C + 1 = 10 + C.
That will mean the 100s digit of RICE must be the same as the 100s digit of the sum.
However, the 100s digit of RICE and LUNCH are different.

Too bad, it was such a cute puzzle!

ALAS + LASS + NO + MORE = CASH

This is the most challenging puzzle from this set.

Some things we notice:

There are ten letters (A C E H L M N O R S) and they must all be distinct.
We are adding three 4-digit numbers and a two digit number to produce another 4 digit number.
A, L, N, M and C are leading digits, so they can't be zeros.
The tens and hundreds digits of CASH (S and A) are also involved in the sums for those digits.

Point 4 has a subtle implication, which I'll illustrate with the hundreds digits. Since L + O must be more than 0, but A is the hundreds digit of the sum, we must have some number of thousands carried over. Because A, L and M are all distinct and larger than 0, the smallest their sum can be is 1+2+3. Putting these two observations together, C must be at least 7.

In this case, I find it helpful to put together a table showing possibilities that we have eliminated:

We can see some more restrictions from the fact that A + L + M must be less than 9. That means we have only the following possible triplets (ignoring order):

{1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 3, 4}

One thing we notice is that 1 is in all of these triplets, so either A, L or M must be 1 and none of the other letters can be 1. Another thing we notice is that we don't yet have any way of differentiating A, L, or M, so any ordering of our triplets is possible. That would mean we have 24 cases to consider.

Let's see how we would work through the cases, starting with A = 1, L = 2, M = 3, the first on our list. Now this, happens to be a stroke of luck, as we'll see.

Starting from the thousands digit, we see that this would make C = 7, if there is a single carry from the hundreds. Indeed, we can see that this must be the value (in the case we are testing), as the carry from there could only come from L + O (plus any carry from the tens digit). Since L is at most 5, L + O is at most 14 and any carry from the tens digit must be less than 6.

Now, in the hundreds digit, we have 2 + O + carry from the tens = 10, so O = 8 - carry from tens.
We know there must be at least one carry from the tens, so O is at most 7. Since 7 is already used by C, let's try 6. That means we need to get 2 hundreds carried over from the tens, so we need
A + N + R + carry from ones = 20, or N + R + carry from ones = 19. Since we have already used 6 and 7, the only way this is possible is if N and R are 8 and 9 (in either order) and we are carrying 2 from the ones.

At this point, the case we've worked through has:
121S + 21SS + 86 + 369E = 71SH

We still have to allocate digits 0, 4, and 5. and we know that S + S + 6 + E = 20 + H. Given our remaining digits, the biggest the left hand can be is if S is 5 and E is 4, making 20. The smallest the right hand can be is if H is 0. Fortunately, this makes the equality hold, so we get our final answer:

1255 + 2155 + 86 + 3694 = 7150

Through the process of checking this case, we learned more about how the carry from lower digits is restricted and it would be faster for us to check through remaining cases.
Let me know how many other solutions you find!

LOL + LOL + LOL + LOL + LOL + LOL + LOL + LOL + LOL + LOL +

LOL + LOL + LOL + LOL + LOL + LOL + LOL + LOL + LOL + LOL + LOL = ROFL

There are 71 LOLs, so this is 71 x LOL = ROFL. While this looks daunting, there are some ideas which take us a long way to the solution.

First, ROFL has 4 digits. If L were 2, 71 x LOL would be more than 14,000, so L must be 1. In fact, ROFL is less than 9861, so LOL is smaller than 9871 / 71 which is 139. We can quickly check

101, 121, and 131 and see that 131 works.

71 x 131 = 9301

Tuesday, February 21, 2017

Cryptarithmetic Puzzles for Grades 1 to 4

Inspired by a series of puzzles from Manan Shah, I decided to have the kids play with cryptarithmetic puzzles today. In addition to borrowing some of Manan's puzzles, I also used some from this puzzle page: Brain Fun. I've included some more comments below about the Brain Fun puzzles.

My main concern was whether the puzzles were at the right level. In particular, I was afraid that the puzzles would be too hard. In fact, I tried solving a bunch of them yesterday and actually found myself struggling. I'll ascribe some of that to being tired and sick. However, my intuition was to make some simpler puzzles of my own. In particular, I added:

puzzles that have many solutions: I figured that many solutions would make it easy to find at least one.
a puzzle that "obviously" has no solution. Now, obviously, the word "obviously" is a sneaky one in math, but I was pretty sure the kids could see the problem with this structure.

Grades 1 and 2

For the younger kids, I started with a shape substitution puzzle. This is one our family explored almost 2 years ago: Shape Substitution. I don't recall the original source.

Two reasons why I started with this. First, it has a lot of solutions, but there is an important insight that unlocks those solutions. Second, by using shapes, we can write possible number solutions inside them as we solve or guess-and-check the puzzle. This made it easier for the kids to see the connection that all squares have the same value, etc.

The second puzzle: BIG + PIG = YUM

Really just a warm-up practicing the rules and doing a little bit of checking that we haven't duplicated any numbers.

The third puzzle: CAT + HAT = BAD

Again, lots of solutions, but noticing leads to a good insight.

Fourth puzzle: SAD + MAD + DAD = SORRY

This is a trick puzzle. The kids know that I like to tease them, so they are aware they need to look out for things like this. We discussed this in class and I suggested they give this puzzle to their parents.

Fifth puzzle: CURRY + RICE = LUNCH

When I translated this to Thai, all the kids laughed. I was sneaking a little bit of English practice into the lesson and then they realized that it was worth trying to read all the puzzles, not just solve them.

Sources: I think I made up all of these puzzles (original authors, please correct me if I'm wrong).

Grades 3 and 4

The older kids already had experience with these puzzles. We did refresh their memory a bit with BIG + PIG = YUM

I asked them to give me the rules and explain why those rules made sense. As with most games, I want to communicate that we're doing things for a reason, but those reasons can be challenged. If they think it makes sense to do it a particular way, we're open to their ideas.

Second puzzle: SAD + MAD + DAD = SORRY

Same discussion as for the younger kids. When prompted, this was pretty easy for them to spot, but they weren't naturally attuned to think about whether a puzzle had solutions or how many. This led me to take a vote on all the puzzles at the end to see who thought the puzzles would have 0, 1 or many solutions.

Third puzzle: ALAS + LASS + NO + MORE = CASH

A puzzle from Brain Fun. I think this is one of the easier ones on that page. Again, a bit of English practice.

Fourth puzzle: LOL + LOL + LOL + .... + LOL = ROFL (71 LOLs)

This was from Manan. I think it is one of the easier ones in his collection, but it looks daunting. Turns out none of the kids in the class were familiar with (English) texting short-hand, so my attempt to be cool fell flat.

Fifth puzzle: CURRY + RICE = LUNCH

Again, everyone was delighted when I translated this one. We're in Thailand, after all, so at least one puzzle had to be about food.

The key exercise

The final assignment everyone (all four grades) was given was to make up a puzzle for me to solve. I was thinking it would be nice to have one in Thai, but we decided to keep it in English as further language practice.

Manan wrote a nice post about having kids design their own puzzles. If it goes well, this is actually the activity that ties a lot of the learning messages together: they think about structure, they think about what allows multiple or single solutions, they apply their own aesthetic judgment, they use their knowledge of the operations, they are empowered with an open-ended task that cannot be "wrong."

We'll see how it goes. At the very least, I expect a lot of work for myself when their puzzles come in!

An extra sweetener

Two kids asked if we could use other operations than addition. That prompted me to put this on the table (also from Brain Fun):

DOS x DOS = CUATRO

Brain Fun Problems

The first time I'd seen the Brain Fun problems, I added them to a list and called them "basic" (see this page.) When I actually went to solve them, though, they didn't seem so easy.

Big confession time: I actually looked at some of the solutions. However, I was disturbed to see that the solutions involved extra information that wasn't included as part of the problem statement! For example, in THREE + THREE + FIVE = ELEVEN, the solution assumes that ELEVEN is divisible by 11. This seems to be the case for several of the puzzles involving written out arithmetic:

TWO + TWENTY = TWELVE + TEN (assume 20 divides TWENTY and 12 divides TWELVE, I wasn't clear about whether any divisibility was assumed for TWO and TEN)

I'm not sure if similar assumptions are allowed/required for any of the others.

Maybe I shouldn't complain, since this assumption creates an additional constraint without which there could be further solutions. Perhaps part of the reason it doesn't sit well is aesthetic. In the 3 + 3 + 5 = 11 puzzle, 3 doesn't divide THREE and 5 doesn't divide FIVE.

Lastly, there is a typo in the final puzzle of the Brain Fun page. That puzzle should be
TEN x TEN = FIFTY + FIFTY

Monday, July 4, 2016

Evens/odds and a quick update

Early years math seems to put a strange emphasis on even and odd numbers. Recently, a friend asked whether there was a point to this. Maybe it is just one of those little bits of terminology that we are asked to memorize for no reason?

By chance, this was something I had started considering about a month ago. It did seem strange that we spend so much time on this simple way of splitting integers. I wondered if it was worth the attention. From that point, my awareness was raised and I started noticing where it occurs and ways it links with more advanced concepts and future learning. My conclusion is that even/odd is surprisingly deep.

First, it is a simple version of concepts that will be developed further. For example, the alternating (starting with 0) even, odd, even,odd, even... is an illustration of a pattern. They will soon see other alternating patterns, then more complicated patterns and 2d or 3d patterns.

For another example, evens are multiples of 2, odds are numbers with a non-zero remainder when dividing by 2. This leads to understanding other multiple families, division, and division with remainder.

Second, the even/odd distinction is helpful for improved understanding of different calculations. For example, the observations that even+even = even, while odd + odd = even, etc. These can be used to help self-check their calculation and also will form early experiences with algebra. Similarly,
even x odd vs odd x odd reinforce understanding of multiplication. Again, this gets broadened for multiples of 3, 4, 5, etc.

Third, there are a lot of more advanced results that are easiest to prove by parity arguments. Sometimes we are working with a set of things that are even and the key observation is we can pair them up. Other times, we have a set that is odd and the key insight is that, when we pair them, one must be left over.

Recently, with the J1 and J2, we were looking at some constrained ways to put the numbers 1 to 25 on a 5x5 checkerboard. They were able to prove that some versions were impossible simply because 25 is odd, so there are more odd integers in 1 to 25 than even integers.

Lastly, there are techniques in computer science that involve even vs odd. This comes up pretty naturally because of the essential use of binary.

Some games

Recently, J1 and J2 have gotten hooked on some classic card games. In particular, we've been playing a lot of 3-handed cribbage. It is a nice way to do some simple addition practice and build intuition about probability. Probability is now getting even more share of mind: in the last couple of days we started playing poker together. This was actually inspired by some of our reading together.

We are reading the Pushcart War.

In one scene, there is a poker game. Of course, the J's insisted that I explain the game and were eager to try it out. J3 was the huge winner tonight, while I busted out. Oh well.

Monday, June 20, 2016

Secret Numbers (Addition Boomerang variants)

We have been enjoying Mathpickle activities in our math games classes recently. This week, the 3rd and 4th graders will be playing with some of the more advanced Addition Boomerang variations.

During our planning, we came up with one extra pointer to tie the activity more closely with multiplication. Also, we had ideas for variations and wanted to record our notes so we can use them again in the future.

Tie with multiplication

In Gord's video explanation, he sometimes records in the center of each loop how many times that loop has been used. We emphasize this and write some related equations to help draw out the connection between the repeated additions in this activity and multiplication.

The first way we do this is by making tally marks inside the loop every time that branch is chosen. At any time, you can pause and write down an equation for the current total in a form

AxN + BxM = Total

where A and B are the values of the loops, N and M are the number of times each loop has been used.

Alternatively, we can show a "completed" round of throws by simply writing the number of passes for each loop in the middle and, again, write out an equation showing the total as the sum of two products.

Secret Number Variations

We start with a basic addition boomerang lay-out, either with 2 or 4 branches, both players (or teams) share a common set of addends and take turns adding on to a common running total. In our variants, the players choose and write down a secret number that helps inform their target for the game:

Version A: players pick a number between 70 and 100. This is their target for the game and they win if the common total hits that value, whether the target is reached on their turn or their opponents turn.
Version B: players pick a number larger than 15. They win if the total hits a multiple of their secret number. For example, if they choose 17 and the running total hits 51 (aka 17 x 3) then they would win. If the total is a multiple of both secret numbers, then the player who chose the larger secret number wins.
Version C: players pick any number. They win if the total hits a multiple of their secret number that is larger than 60 (not equal to 60). If the total is a multiple of both secret numbers, then the player who chose the larger secret number wins.

Version B is, I think, the most directly playable.

Possible issues
I'm not sure how to deal with the case where both players choose the same secret number.

For Version A, it will be interesting to see what modification kids can find that will deal with the fact that it is very easy to miss any particular target. In the basic game, once the total is larger than your target, there is no hope of recovery. There are several ways to address this. I would be eager to hear any rule sets that kids create and hear about the experiences.

In Version C, I wonder if choosing 2 as the secret number is too strong a move?

Monday, June 6, 2016

Broken Ruler and Multiplication refresh

Ruler Explorations

We noticed that one of our tools, a ruler, had gotten broken.

Is it still useful? As a challenge, J2 looked at measuring a noodle from his soup.

There were two ideas:

the noodles were too long, so had to be broken in pieces to measure with the remaining ruler
Our ruler doesn't have to start at 0, we can use subtraction!

While we were talking about this, I recalled the idea of Golomb Rulers. We came up with a ruler that was marked only with 0, 1, 3, 7, 11, 12 cm. This lets us measure 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12cm distances.

What if this ruler gets broken? For example, we imagined cutting our ruler between the 1cm and 3cm markings. What measurements are still possible? Is there anything interesting in the relationships between how many ways there were to measure a distance before the break and how many ways to measure after?

Multiplication Refresh

We recently re-watched Graham Fletcher's Progression of Multiplication. Both J1 and J2 did some practice around this. The most interesting point was J2's reaction to Graham's comment at 4:54: "This sucks!"

"Why did kid's say that?" "Hmm, let's try out a couple of examples..."

We rolled dice to randomly generate digits for an example and were lucky to get 35 x 34. J2 quickly saw this as 35 x 35 - 35 and knows a pattern that let him quickly calculate 35 x 35 = 1225. As a result, 35 x 34 was pretty easy for him to calculate.

Then, he worked through a graphical representation and a powers-of-ten version. At the end, we got to compare and contrast the different approaches.

Continuing to play with some old activities

Fold-and-punch
We did some more fold-and-punch activities. This time, we folded the paper, then drew a location for the punch, and tried to figure out how many holes would result and where they would be. We broke out our serious hole-puncher:

Unfortunately, must be operated by an adult

In this example, we got a small surprise that the result wasn't a power of 2:

Chairs (and tables)
Another round of building chairs, following the NRICH activity. This time, with J3:

We got to compare and contrast our designs:

how many cubes were used for the legs? Which one had more and how many more?
How many cubes were used for the whole chair? How did they compare?

Monday, May 30, 2016

Class summaries

Quick notes on the activities for the classes today.

Grades 1 and 2

Mobius loops
We are splitting up games with other activities. This week, we are introducing several geometry explorations. First up are Mobius strip activities, nicely shown in this Matt Parker second favorite shape:

cylindrical loop: using 2 colored pencils, draw a line along the center of the loop on the inside and the outside. We can see that there are two sides, no big surprise. Cut along one of the lines and the loop splits into two new loops
Mobius strip: give the paper a twist. Now, draw along the middle of the paper and see that there is only one side. Now, cut along this line and see what happens. Repeat this, drawing another line along the center of the new strip. Do you have one side or two? Again, cut along the central line. What do you get?
Two connected loops: tape two, untwisted loops, together in perpendicular directions. Now cut along the center lines of each loop. What do you get?
Two connected Mobius strips: tape together two mobius strips and cut along their center lines. What results now? Did everyone get the same result?

There are some natural extension explorations:

try these with more twists (as per Matt's video)
keep cutting the center lines
Connect a Mobius strip and an untwisted loop (half-way step between 3 and 4). Now, cut along the center lines. What happens?
Inspired by the thinner and thinner loops, kids can explore ways to cut paper so that they get longer and longer strips or loops

Note: these activities can be even more rewarding when something goes wrong. For example, what if there isn't enough tape connected the ends of the loops? These mishaps make everyone pause and consider more carefully what is actually happening.

Also, in the class, we only had time for the first two make-and-cuts, then demonstrated the two connected loops.

Punch (fold and cut)
All this cutting fits nicely with our second exploration: the punch activities from Joel David Hamkins' post punch, fold, and cut from Joel David Hamkins.

Grades 3 and 4

We started with the Shapes x Shapes puzzle from NRICH:

We added a couple of extra questions to this challenge:

Before completing the puzzle, which numbers do they think are excluded? Why?
Make extra equations that allow us to include those missing numbers. Are they easier to incorporate using multiplication or addition equations? What about equations that combine multiplication and addition?

Observations:

Once again, this appears to be a very simple activity, but gave us a lot to talk about. In particular, it was very helpful for highlighting a lot of misconceptions and gaps in understanding. Examples:

"identity" relationships were still unclear: 1 x n = n, 0 x n = 0
Several students thought the first equation would be 4x4x4 = 12 (confusing multiplication and addition)

Division Dice move to Cards
For our core activity, we are extending the Division Dice game. This time, we use playing cards, A through 10, instead of dice rolls to generate the random components of their equations. In this case, the aces are wild and can be any number from 2 to 10. When they form a multi=digit number with a 10, the 10 counts as two digits. For example, 3, 5, 10 could form 105 ÷ 3.

With cards instead of dice, we lack the natural move of flipping the dice to the opposite side which we used to make sure all throws could give us whole number divisions. In this version, we allow division with remainder. However, the twist is that the remainder becomes points for the opponent.
For example, if I draw 3, 3, 5, I can form 53 ÷ 3 to score 17 points for myself, but the opponent gets 2 points.

Question: Are there cases where the best play is not to form the largest possible number divided by the smallest number?

An interesting game variant: swap the scoring so that the active player scores the remainder and their opponent scores the quotient.

Improv Math and Division Dice follow-up

We had a really good experience playing Division Dice, the game that we introduced a couple of posts ago. Mainly, I want to illustrate something fun that came out of really listening and paying attention to what the kids are doing and saying. I like to think of this as "improv math," as a way to credit my improv comedy experiences for heightening my awareness of how important this is.

Division Dice for number sense

I was really pleased about the quality of thinking stimulated by the game. We played with the most loose rules (1s are wild, the components of the 2 digit value can be flipped to their 7s complement). That gave a lot of opportunity for the kids to think through options to (a) make whole number divisions and (b) maximize values.

For example, rolling 3, 4, 6:

what are the allowed groupings that give a whole number division? Remember, in the 2 digit number, we can use any of the values 1, 3, 4, 6, and it is possible for us to use two 3s or two 4s in our calculation.
What is the highest scoring choice?

Division Dice for arithmetic exercises

As a way to create virtual worksheets, this game is mediocre. The basic structure means that students are never dividing by a divisor larger than 6. This leave out a lot of fact families. However, because the kids are trying to maximize their scores, they quickly realize that they can almost always get away with division by 2, occasionally must divide by 3, and rarely get stuck dividing by 4 or 5. I haven't yet seen a case in a live game where division by 6 was necessary.

Fun exploration: what scenarios will require division by 6?

Using playing cards or other dice shapes allows us to extend the possible values and reduce the likelihood of dividing by 2 or 3. However, it also increases the number of cases that don't have a whole number division relationship. We are thinking about ways to incorporate division with remainder and will try out a variant tomorrow.

Improv Extension

Playing at home, the 3, 4, 6, case led J1 to consider: how do 63 ÷ 3 and 64 ÷ 4 compare?
As he contemplated that, I realized that we had a nice sequence of multiples, meaning all of these are whole numbers:

There were several cool things for J1 to observe here:

4 of the 6 quotients end in 1
The quotients are all decreasing
The drops between successive quotients are themselves decreasing
the dividends are equal to the divisors + 60

We pursued this in two ways:
Extension 1: what if we add something else to the dividends?
We tried three versions.

starting with 60 and adding 6 at each step
Starting with 60 and adding 60 at each step.
starting wit 1 and adding 7 at each step

You can see our notes mid-discussion below:

Later, when J2 was also involved, I offered them another sequence: starting with 66 and adding 6 for each increment:

66 ÷ 1

72 ÷ 2

78 ÷ 3

84 ÷ 4

90 ÷ 5

96 ÷ 6

120 ÷ 10

132 ÷ 12

150 ÷ 15

180 ÷ 20

240 ÷ 30

420 ÷ 60

3660 ÷ 600

36060 ÷ 6000

We're breaking the rule about the dividends being multiples of the divisors, but the last two calculations are still easy and nicely illustrate the limiting behavior.

Extension 2: can we find other chains of whole number division equations?
We started this by thinking more simply: for chains shorter than 6. For example, what are the smallest K, L, M, N larger than 1 such that all of the following are whole numbers:

K ÷ 1

(K+1) ÷ 2

L ÷ 1

(L+1) ÷ 2

(L+2) ÷ 3

M ÷ 1

(M+1) ÷ 2

(M+2) ÷ 3

(M+3) ÷ 4

N ÷ 1

(N+1) ÷ 2

(N+2) ÷ 3

(N+3) ÷ 4

(N+4) ÷ 5

After getting the shorter cases under our belt, we then went for a chain of length 7. J2 worked by himself for a while, then came back and announced that no chain with dividends smaller than 100 would work. He went away and then came back quickly with the idea that maybe we could add 7! to each divisor.

Tuesday, May 24, 2016

Logic Puzzle collection and Dropping Phones

Recently, Mathbabe put out a request for riddles. There are some good links in the comments:

The Riddler on FiveThirtyEight
Logic Puzzles on Math is Fun
Alex Bellos at The Guardian
Riddles.com (naturally)
Puzzling.stackexchange.com (is there a stack exchange for everything now?!)
Authors cited: Dudeney, Gardner, Polya, Smullyan, Vaderlind.

We've played with puzzles from almost all of these sources in the past, but this was a good opportunity to put together a nice list.

The one that stood out to me was on FiveThirtyEight. That's a site I read frequently, especially during this US election season ... and I was totally unaware of The Riddler feature. We kicked off with the oldest puzzle from their archive: Best way to drop a smartphone.

Breaking stuff

Right away, J1 and J2 loved the theme and were into questions about whether they could somehow keep the phones, if they weren't broken, or utterly destroy them, if they were broken. We talked about setting an upper bound with a very simple strategy of starting on the 1st floor and working up each floor. That's not a great answer, but it got them into modifications and improving strategies.

Through the conversation, it was interesting to see them start with the idea that the drops for one of the phones would be de minimis and could be ignored, but then start to pay attention to that aspect. Also, they had to grapple with the idea of balancing the number of drops that would be required in different cases.

While we didn't get to the optimal strategy, but the kids managed to get a version that, at worst, would take 19 drops for the 100 story building. Their intuition was based around taking the square root of 100. They could see that this probably wasn't the best answer, since there were still cases that, at worst, would take 10 drops and others that, at worst, would take 19.

Can you do better?

Smaller before bigger

The puzzle page poses the same challenge for a 1000 story building. However, we found something interesting when working on the version for a 10 story building: there is strategy where all worst cases take the same number of drops, but it is not the optimal strategy!

It is a little hard to write about this without disclosing the strategy, but here's a hint:

4 + 3 + 2 + 1 = 10 and 5 + 4 = 9
We are always allowed to assume that the phones will break if dropped from the top floor of the building

This led to another extension: what size buildings will have the same issues as for a 10 story building?

Probability comes in

Another extension is to think about the expected number of drops required and strategies that minimize this. Crucially, this extension introduces the idea about our prior beliefs about the sturdiness of the phones: where do we think the phones are likely to break, what is our confidence?

We didn't pursue this extension very far, but it did lead to some interesting conversations about terminal velocities. For those who want to follow that thread, this (other) stack exchange thread might suit you: How to figure out height to achieve terminal velocity.

Monday, May 23, 2016

Division Dice (math games class)

Who: grades 3 and 4
Where: in school

A dice division game

We created a simple dice game to practice division. Here's a description of the basic element of play:

roll three dice: for example, 3, 4, 5
group two of them into a two digit number: for example, 45
Divide the two digit number by the remaining single digit: for example, 45 / 3 = 9
This value is your score for the round
First player to 200 or more points wins (we used 100 for the initial game)

Key constraints

You can only score points if the single digit is a factor of the two digit number (remainder must be 0)
Where there are multiple options, the player can choose the combination that gives them the maximum score

This pencilcode program (see code) analyzes this basic game structure, identifying how often there will be no legal scoring arrangement and showing a histogram of the largest scores.

Modifications/Extensions

I wasn't satisfied with three elements of this game: (a) any time a 1 occurs, the division calculation is too easy, (b) too many combinations don't allow a score (about 15%) and (c) there aren't many decisions for the students to make (just six combinations to investigate).

We addressed these by adding two extra rules:

1 is a wild that must be replaced by a value from 2 to 6 (cannot be left as a 1)
On your turn, you can flip the over the dice in the two digit number. For example, a 6 can be flipped to a 1, 5 to 2, 4 to 3, etc.

The first point removes the division by 1 cases, the second one allows more choices and reduces the number of non-scoring cases.

*UPDATE*
Allowing the dice flip and wild 1s seems to make the game too loose. Instead we dropped the wild 1s rule and added these two:

Division by 1 is not allowed in the game
If you roll triple 1, re-roll

Sunday, May 15, 2016

Hive at the beach (quickie)

Have been at the beach for the last couple of days before school starts. We got to play with one of my recent impulse buys: Hive (carbon).

Chance for discovery

As an experiment, I started by laying out the pieces with the blank sides up. One at a time, I asked the J's to come in and tell me what they noticed. For each of them, at some point, there was a moment when they tipped or turned a piece over and discovered the insects. Their reactions were a real delight and they realized it was special feeling, so quickly helped rearrange the pieces and get another sibling so that they could have the same experience.

Making patterns

Having started with the blank sides of the pieces, it felt natural to the kids to sometimes play with these tiles just to make patterns. Here is one example:

Playing the game

Of course, there is also delight in playing the game itself:

From this detail, you can see that J3 (playing black pieces) is following an unorthodox strategy. At this stage, she is really learning the rules and sees the whole activity as a strange way to play together to create unusual patterns and combinations of the bugs:

Sunday, May 1, 2016

Reversed inequality

Recently, Mike Lawler posted a challenge base on the first question from the recent European Girls Math Olympiad (side note: what is "European" about this contest now that a US team participates?)

I enjoyed thinking about this problem and wanted to come up with something related to do with our kids. One way to see this problem is that it appears to reverse the direction of the inequality between the arithmetic mean and geometric mean. My version for younger students involves exploring this inequality first.

Discovering the Arithmetic Mean-Geometric Mean Inequality

I had J1 and J2 select two dice from our pound-o-dice. Initially, J1 chose a d20 and d6 while J2 chose a d12 and d6. I asked them to make a table with 6 columns and 7 rows (we ended up adding more, so probably better to ask for 8 columns and 10 rows). I had them label the columns:

A
B
AxA + BxB
2xAxB

They rolled their two dice and put the values into columns A and B, the calculated the other two columns as labeled.

J1's first results were 16 and 5. After filling in the rest of that row, he paused and then switched to 2d4. This wasn't a problem for our investigation, but he did miss some calculating practice with more difficult seed numbers.

After filling in 6 rows of data, I asked them what they noticed.

Here were some of the observations:

when A and B are the same, our calculated values are the same
when A and B are different, our calculated values are different
When A and B differ by 1, the calculated values differ by 1 (also vice versa)
AxA + BxB > 2xAxB
We are only using positive integers

J2 really got into the spirit and asked for some more suggested columns. I told him to add:

AxA - BxB and (A-B) x (A-B). J1 also added the square of the difference.

With that, they noticed two more things:

AxA + BxB was larger than (A-B)x (A-B)
AxA + Bx B = 2x Ax B + (A-B)x(A-B)

You can see that they also added some negative numbers on J1's sheet and challenged one of these observations.

A picture proof

To finalize our exploration of the inequality, we used tiles to build intuition for a picture proof of this identity:

AxA + Bx B = 2x Ax B + (A-B)x(A-B)

This is one of the pictures we liked the most. In this case, A = 5 and B = 2. Along the left side and the bottom are rectangles AxB (the green + yellow and the blue + yellow regions). These two rectangles overlap in a BxB square (the yellow region) which they also highlighted with the 4-color square. The remaining red square is (A-B)x(A-B).

Side note: In our discussion, I was delighted that J2 would correct me whenever I slipped and said "greater than" instead of "greater than or equal to" when discussing the inequality.

Two quick algebra extensions

For kids who have already done a little algebra, proving the identity using the distributive law should be fairly straightforward. The other little extension is to show that our expression A²+B²≥ 2AB is equivalent to the arithmetic mean-geometric mean inequality.

Friday, April 29, 2016

4 Coins challenge

Recently, we were at lunch and the kids asked for a challenge to entertain themselves. I had just read this Numberplay column and had Lora Saarnio's Four Coins Problem in mind.

The basic rules are to choose values for coins in a new system so that any value from 1 cent to 10 cents can be made with a single coin or with two coins.

For our older two, this was a great on-the-go puzzle without pen and paper. Later, we got into some extension questions that required more careful organization of their observations and attempts.

Mostly, I want to report some of the extension questions that we found interesting. The key point: this problem is very accessible (J3 could also work on a simplified version) but it is ripe for a Notice & Wonder discussion that will reveal increasingly complicated and challenging extensions.

Note: we used names of neighboring countries, but none of the claims in these scenarios are true (as far as we know).

Number fussiness
In Burma, the president really likes the sequence 1, 2, 3. Is there a coin system we can suggest that includes coins with all three of those values?

No duplicates
We noticed that, for all the coin systems we created, there were always some values that were duplicated. What we mean is that there are two ways of using the coins to make that value. For example, in the coin system 1345, the following values can be made two ways:

4 = 4 = 1+3
5 = 5 = 1+4
6 = 3+3 = 1+5
8 = 4+4 = 3+5

Is there a coin system where all the values can be made only one way? If not, why not?

Two rival countries
Cambodia and Laos really love to compete with each other. As a result, they want coin systems where there aren't any shared coin values. For example, if Cambodia has a coin with value 5, then Laos can't have a coin value 5.

Is it possible? If not, what is the smallest overlap possible and how many possible coin systems achieve this smallest overlap?

Note: when we started this investigation, we had already designed a new system for Cambodia, so our starting question was whether there was a new system for Laos that wouldn't overlap this particular system for Cambodia, how much overlap was unavoidable, and then how to change both systems to minimize the overlap.

More values!
If we want to extend to amounts up to 20, how many coins are needed to have a compliant system (any value possible with only 1 or 2 coins)?

We haven't pursued this to a final answer, but the J's quickly recognized that, for any compliant coin system covering up to 10, they could create an 8 coin system that would cover up to 20. However, they also realized that four coins would be too few. This means that the minimum required to cover up to 20 is 5, 6, 7, or 8.

Coin triples
In the More Values! extension, we allow more than four coins in the system. The other way to relax the original constraints is to allow more than 2 coins to build a value. What can we achieve if we allow three coins at a time? Could we use fewer than four values in the system and still cover all amounts from 1 to 10? What about 1 to 20?

Tuesday, April 19, 2016

Tienanmen Attempts (Gord! festival continues)

We have been continuing the Gord! (BGG entry, Math Pickle site) games festival that started when we learned about Santorini. Today, the game we started learning is Tiananmen.

Game theme
While I have told the Js the name of the game and the designation of the sides as police/communist party and students, we haven't talked about the events of 1989. If you have a strong opinion about the game theme, feel free to let me know in the comments.

For now, this isn't our focus.

Creating a Board

This game is played with a go/baduk/weiqi set. While we recently got excited about that game, too, we haven't yet purchased a set. That means, once again, part of our efforts for this game are DIY/substituted versions.

Note: while the rules are very simple, we initially failed to understand how to make a legal move, thinking that we were able to play anywhere that was connected (including diagonally) to the batch of "stones" played by the previous player. That variation makes it far too easy for the students to win. In most of the pictures below, there will probably be a bunch of illegal positions.

As a first attempt, we played checkers vs dinosaurs (+mammoths) on a chess board:

For this version, we shrank the target in the center to one intersection. Also, you can see that we were playing on the spaces, not the intersections of the board:

In this version of the game, it was far too easy for the students to win.

For our second attempt, we used square tiles to recreate a full sized board. Unfortunately, this is how many yellow, green, blue, and red tiles we had:

That made for a nice little exercise: how many tiles are missing from the square?

Not so nice looking but we did fill in the rest of the board with other square tiles (the whites are from our 100 board, so this gives another way of quickly seeing how many were missing). As you can see, the TRIO set again came in as a central monument:

On the larger board, we used blue 1 cc cubes for the police and wooden 1 cubic inch cubes for the students. Eventually, we ran out of wooden cubes and had to substitute circular magnets and then colored tiles.

If you look carefully, you can see that we still failed to understand the rules at this point: playing on the squares rather than intersections and making illegal moves.

An epilogue and recommendation
Finally, we did figure out how to play. For those starting out, especially those without a go set, I strongly suggest playing first on the intersections from a chess board (9x9 grid) with a monument in the central intersection. This is a version that is easily accessible for young kids, but still has connections to the strategic thinking of the larger game.

Also, you don't have to lay out 324 tiles in a nice arrangement!

Tuesday, April 12, 2016

Powers of a permutation and Santorini review

Two unrelated topics today. Or ... I guess one could argue that everything in math is related, but I don't see direct connections myself. If you spot some, let me know in the comments.

Rainbow permutations

We have a collection of crayons that can stack. Well, to be honest, mostly the kids break the tips off. The second most common use is to stick them on fingers as fancy fingernails. The fourth most common use is to actually draw or color with them.

J2 was engaged in the third most common use, stacking, when he noticed something new. He started with the colors stacked in rainbow order (R O Y G B Purple Pink). Then, he took 2 off the bottom and moved them to the top. Then, he took the bottom two and moved to the top and repeated. He noticed that, eventually, he got back to the starting order. Experimenting further, he tried the same process moving 3 from the bottom and repeating. Again, he eventually got back to the starting order.

Here is an example of the crayons stacked together:

In this case, he is moving blocks of 5:

After showing me, he suggested trying 4, 5, 6, then 1. He noticed a couple of things:

moving 6 is like moving 1 backwards (from the top to the bottom).
Similarly, 5 and 2 are related, 4 and 3
7 is prime, so maybe that is the reason the arrangement repeats

To test his hypothesis, he added another crayon (for 8) and tested. Again, he got back to the original arrangement. Hmm, doesn't need to be a prime!

Where should we take this first foray into group theory?

Santorini

Gordon Hamilton of Mathpickle, one of our favorite game and puzzle resources, has a Kickstarter for the newest version of his game, Santorini.

I encourage you to check it out. (For what it is usual disclaimer applies, I'm not financially related to this game or producers in any way.)

For reference, this is what the previous version of the game looks like:

Our DIY board (version 1)
While this version of the game will have nice custom pieces, the underlying game can be played with almost any collection of stackable objects. For our introduction to the game, we tried using our TRIO blocks:

To explain what you're seeing:

1x1x1 cubes are used for building levels. I originally thought we would color coordinate (red for first level, pink second, etc), but J1 and J2 liked mixing up colors.
4 unit straight connectors for our builders: magenta versus green.
Angle or arc connectors are radio antenae to serve as the fourth building level and block further building
Flags and wings as boundaries of the 5x5 board

Reactions

Both older Js enjoyed the game play. We quickly played 5 or 6 times. This game clearly has a lot of depth. We will have to play a lot more to see what patterns we can identify and whether we can develop any opening strategies. They are also very eager to play with some god powers. Perfect way to spend the rest of the holiday this week!

The TRIO blocks have pros and cons for this game. One of the best features is that it makes the game set-up robust to tipping the board, knocking the pieces off, or otherwise unsettling the position. It was clear to see the sizes of the levels and also immediate to identify towers that had already gotten killed (with a fourth level antenna).

In our play, there were two drawbacks. First, the straight connectors are a bit hard to remove from their positions. Second, the higher towers (2 and 3 levels) sometimes obscured allowed diagonal moves in a way that the stacking tiles version didn't. I wonder if this will also be the case with the new version of the game as the building levels seem tall relative to the size of the builders.

Longer-term, I wonder about buying customized games vs playing with generic materials. It is easy for the kids to grab a box off the shelf and start playing. Somehow, I suspect they will be less likely to grab a building set and a notecard with rules.

More thoughts on DIY game versions

Before Santorini, I had gotten excited about making DIY versions of the GIPF project games. I was thinking:

these games will be fun to play, so great motivation to replicate them
interesting challenge to make our own triangular grid board (squares we already have in abundance)
good creativity prompt as we repurpose items
it would make the kids feel power over the game structure and rules, leading to exploration of variations and deeper thinking about structure.

However, the kids, particularly J1, were surprisingly unenthusiastic. With Santorini, again, they didn't take much initiative in developing our DIY version. However, I wonder if they will be more motivated to modify or replace the TRIO blocks board since they now see that the game is fun and have experienced some of the limitations of our current version.

On the other hand, maybe they'll just push me to buy the commercial version.

Monday, March 7, 2016

Assorted math conversations

A collection of little items from the last month.

Optimal time to eat

P was talking with the kids about what order they should eat their food, which led to lots of graphs showing flavor over time. I don't know if this is a typical conversation for other people, but pretty common in our house. Here are some of the charts they made: Naturally, this also led to other charts. In particular, a debate about how quickly someone could develop skills by practicing music. Note the lack of labeling on the y-axis.

Some challenges

At lunch today, everyone got a math question. Apologies, I don't remember what challenge we gave to J1. J3 was given what is 5 + 2? After we let her consider and answer, J2 got asked what is the cube root of 125? Before either older kid could answer, J3 yelled out: 5.
Hmm. No, we also don't think she really understood, but it is nice to dream!

A joke

J2: look, a prime! (pointing generally at the 100 chart on our wall)
Everyone look expectantly at the wall
J2: No, it isn't. 22 is a compositive
All little Js burst into hysterical laughter. P looks at me as if to say, "this is your fault."

I wonder

From nowhere: Daddy, what 3 digit number has the most factors?
Exploration to follow tomorrow.

Cooking

J3 and I made scrambled eggs today in two batches. First, with milk for J1, using 2 eggs. Then, a second batch without milk for J3 using another 2 eggs.

J0: how many eggs did we use
J3: For J1, we used 2 (holds up two fingers on right hand). Then, we used two for me (holds up two on left hand). Let's count them . . . 1, 2, 3, 4. Four eggs!

Contortionist cubes

We got a lot out of this video from Mathologer: Contortionist cubes.
An immediate idea was to see which of the objects we could construct using our materials at home. The best we managed was using polydron to build one of the contortionist cubes, with sides that are either all red or all green. This high-tech enterprise was made possible by scotch tape!

Thursday, February 18, 2016

How to do science (short story)

J1 has been at home sick the whole week. Each day, he writes a journal entry, usually a short story.

How do people jump?

One day, Ms. Jump got a call from Ms. Silly. She asked her to go to her home to jump for her
because she was doing research on how to jump.

When Ms. Jump got to Ms. Silly's home, she gave Ms. Jump 50 pounds as payment.
So Ms. Jump jumped and Ms. Silly recorded the jump.

Ms. Silly said, "Oh, now I understand. We make our head go up and down to jump!"

Tuesday, February 16, 2016

Dots and Boxes variation

In grades 1, 2, 3, we played this variation of dots & boxes: Mathify the Squares Game.

I'm enthusiastic about this game, but can't resist a quick comment about the "mathification." Dots & boxes is already a mathematical activity, it doesn't need to be "mathified." This term implies confusing arithmetic and calculation with math, something I've written about elsewhere and, I hope, is clearly not implied by our blog.

In any case, I'll use the shorthand MD&B to refer to this dots and boxes variation.

Notes from playing in class

In class, we first introduced the kids to vanilla Dots & Boxes with a pre-printed grid of dots. We knew that it would be too much to play on a lattice covering the whole A4 sheet, but we thought a quarter of the grid would work. That turned out to be too big and the game started to seem monotonous to the kids as there was too much time spent on the opening (playing on squares that don't yet have any filled sides.)

We rectified this problem in 3rd grade and played on much smaller grids, with sizes between 7x7 lattices (which yield 6x6 squares) and 10x10.

After they were comfortable with the vanilla game, we introduced the product version with dice. In our case, we just had the players take turns and didn't give an extra turn when someone completed a square.

Some rule variations
There are some simple variations depending on how you deal with completed squares:

no extra turn (this was the version we played in class)
player adds another side to a different square with the same value. For example, say the dice are 2 and 4 (product 8) and the player fills a square. They also must add a side to another square with value 8.
player rolls the dice and adds a new side (a full extra turn)
player adds a side to any square (dice and square values ignored)

Of course, you could also make the extra turn optional instead of compulsory. You might also have some ideas about different ways to handle cases where there are no more free sides on squares with the required number value.

Probability questions

Dice games naturally lead to probability questions. Here were two that I really liked, based on scenarios from a recent game play:

What question are we asking?

What is the chance a player will get both the 20 and 36 boxes:

One great answer was 1/2. The reasoning: we are going to play until all boxes are filled and each of us have an equal chance to fill this box, so 1/2. This is not quite right, since the person who is about to roll has an advantage, but I thought it was an interesting interpretation of the question.

A 2x2 square

In this configuration, what is the probability that the next throw will allow the player to complete a box in the 4-30-25-15 zone?

A bigger D&B family

One of the reasons I thought this D&B variation was so cool is because of our games matrix. Whenever we play games, J1 and I talk about some key characteristics of the game, particularly the amount of randomness and the strategic complexity. These are not entirely independent dimensions, since a larger amount of randomness reduces the number or importance of each player decision, thus the strategic depth.

The mechanics of this game gave us some ideas about how to dial up or down the amount of randomness in this family of games. Here is our list of members of this family, roughly ordered from least random to most random:

vanilla dots and boxes: no random element
mash-up with product game: squares are still labeled, but players control the two factors using selectors (like in the product game) instead of using dice. This can be played with different collections of factors and different size boards (including board variations where a product appears multiple times or only a single time, where values are ordered or randomly distributed).
half-way house: one factor is chosen by players moving a selector, the other is determined by a dice roll (either before or after the "free" factor is selected).
MD&B game: as played in class and described in the first link
MD&B game where each factor appears only once. This is a case of dialing up the randomness by reducing the strategic options of the players.

Ideas for other games

I'm excited to see what other games we can modify use the underlying idea from the MD&B variation. To be clear: use numbers to label parts of the game and then constrain the players' actions based on a die roll to involve either the pieces with corresponding labels or board positions with those labels.

Three specific examples:

Hackenbush variation where segments of the picture are numbered. This could nicely incorporate probabilities by putting values of the least likely dice rolls closer to the ground.
Ultimate tic-tac-toe meets the product game: from Art of Math. This is an old post, but I just happened to see it when preparing this post.
Dice chess. Here's the wikipedia article. For some reason, I often forget about this variation, even though it is a nice way to reduce the strategic complexity of vanilla chess for beginners and has some nice links with probability.

If you have some favorites, I would love to hear in the comments!

*Update*
Playing through the MD&B version several times, we came up with these rule variations that are worth your consideration:

Game stops as soon as someone rolls a value that can't be played (alternatives are to let that person roll again or have them pass their turn)
Remove some of the randomness: (a) on your turn, you roll and play a side with the required value, but they opponent also plays a side with that value. As we played it, that means moves (without filled boxes) go: A, B, B, A, A, B, B, etc. (b) When a player fills a box, they can choose to re-roll both, either, or none of the dice for their extra turn.

I think our favorite was a combination of all three of these components. Mixing 2a and b, you have to be careful to keep track of whose turn it is, but it lead helped bring out elements of strategy and more thinking about probability.

*Update 2*
Game phases
Above, we talked about how A4 (or even 1/4 of an A4, which I guess is equivalent to A6) is too big for beginning players of Dots & Boxes. They found the game "boring." J1 and I talked about this experience and it led us to considerations around game phases: opening, middle game, and end game. These are terms we first learned in chess, and we found it useful to contrast the two games.
Here were some observations:

Opening: a lot of choices, not obvious how most of those choices link with "scoring" or the winning objective of the game. At this stage, there seems to be little interaction between the players (there is enough space that most of their actions either don't bring opposing pieces together or there is a lot of open territory).
Middle game: still many playing choices, increasingly direct conflict between players, interim objectives within the game become more clear and there are some chances for plays that either score or more clearly move closer to the overall game objective.
End game: significantly fewer choices for each turn than the other phases, either because there are fewer pieces (chess) or most territory has been claimed. At this stage, players are able to focus on the overall game objective, rather than interim objectives.

What we realized is that the larger playing area for D&B significantly increases the length of the opening. Because this phase is the least connected with capturing boxes, it is the hardest for beginning players to see how their choices ultimate lead to scoring and it is the phase with the most available choices on a turn.