Wednesday, March 29, 2017

Some simple dice games

J3 and I played several simple games recently that I want to record. One of them, race to the top, is a variation of a more sophisticated game that can be used more generally.

Digits three in a row

Materials: a 100-board, 2d10, colored tiles
Players: 2
Goal: mark three spaces in a row
Basic play:
Two players take turns rolling the two dice. On a player's turn, they form a 2 digit number using the two digits and then claim that space on the 100 board. If the space has previously been claimed, they lose their turn.

The first player to claim three adjacent spaces in a line (horizontal, vertical, or diagonal) is the winner.

(1) we used one dice marked 0-9 and another marked 00 - 90 (all multiples of 10) and then added the values to get our two digit numbers. This eliminated any player choice, but helped reinforce the idea that the value in the 10's digit is the number of tens.

(2) the winning condition can be increased to require a line of 4 spaces

(3) the winning condition can be changed to allow any three (or four) spaces that are colinear; these spaces would not have to be adjacent.

This is a very simple game, especially the variation we played, but J3 found it fun. It was a useful exercise to practice locating the numbers on the 100 board.

Race to the top

Materials: An 11x 6 grid with one side labelled 2 to 12, 4d6, 3 tokens. Optional: 11 distinct tokens/small objects per player.
Players: 2 - 4
Goal: capture 3 columns.
Basic play:
This game is fairly simple, but has resisted our attempts to succinctly summarize the rules. Here is an explanation as we play two turns.

Here's our playing material, the three green squares are temporary markers:

First player, J1 rolls two ones and two fours. With this roll, J1 could group them into two fives or a two and an eight:

J1 decides on two fives and puts a temporary marker on the second level of the 5's column. After you understand the rules consider whether this choice is better or worse than the 2 and 8.

J1 chooses to continue rolling and gets 1, 1, 4, and 6. J1 groups these as a 5 and a 7, then moves the temporary marker in the 5's column up one level and adds a marker at the bottom of the 7's column.

J1 rolls a third time, getting 1, 1, 2, and 6. The only option is to group these as 3 and 7, so J1 places a temporary marker in the 3's column and advances the marker in the 7's column.

At this point, J1 ends his turn and marks his progress. On his next turn, if he gets a 5, for example, the temporary marker will start at the fourth level of the 5's column (building on his consolidated progress).

D has the next turn. He gets 1, 4, 4, and 5, which he chooses to group as 5 and 9:

D chooses to roll again, getting 4, 5, 5, and 6. This has to be grouped as 9 (advancing in that column) and 11:

D chooses to roll again and gets 2, 3, 5, and 6. This is a lucky roll that can be grouped as 5 and 11, allowing two tokens to advance:

D presses his luck and rolls a fourth time, getting 1, 3, 3 and 5. The dice can't be paired to get a 5, 9 or 11 and there are no more temporary markers available to place, so D loses his progress. J1 will have the next turn.

To be clear about the failure condition: the player must be able to place or advance a temporary token for both pairs of dice. For example, if D had rolled 1, 3, 3, and 6, he still would have lost his progress.

Further rules:

  • the first person to end their turn on the sixth level of a column "claims" that column.
  • the first person to claim 3 columns wins the game.
  • columns that have been claimed by any player are safe values for all players. Players do not need to allocate a temporary token to those columns.
  • Players can occupy a square that an other player has marked.

(1) Change the winning condition so that the first player to capture a column wins
(2) Change the height of the columns, either fewer than 6 for a faster game or more than 6 for a slower game
(3) Change the failure condition so that only one pair of dice needs to be playable and reduce the temporary tokens to 2.
(4) only allow each player to roll one time. This eliminates the "press-your-luck" aspect of the game and is much more basic.
(5) allow players to jump over a square that has been occupied by another player. This rule particularly fits well if you use objects to record your consolidated progress (which also makes the grid re-usable).

I was originally taught this game by Mark Nowacki of Logic Mills.
J3 and I played the variation where each player only rolled one time on their turn.

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.

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:

  1. B, G, I, M, P, U, Y must all be distinct
  2. We are adding two three digit numbers and the sum is a three digit number
  3. B, P, and Y are all leading digits
  4. 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!

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

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
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!

This is the most challenging puzzle from this set.

Some things we notice:
  1. There are ten letters (A C E H L M N O R S) and they must all be distinct.
  2. We are adding three 4-digit numbers and a two digit number to produce another 4 digit number.
  3. A, L, N, M and C are leading digits, so they can't be zeros.
  4. 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!


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

Thursday, February 23, 2017

A bit of 3D(ish) geometry

Note: this post started out focused on two recent geometry projects. However, the Desmos Function Carnival, which I originally just included as a miscellaneous item, is also worth your time.

Nets and solids

The most ambitious project recently was led by P (the mom, of course). She found nets for 3-d shapes and supervised J1 and J2 as they created a nice display to take to school.

Here are the nets: SenTeacher Polyhedral nets. They have a collection of other printables, but this collection of nets seems to be the most interesting. Take a look and let me know if you see anything else worthwhile.

Here's the completed board:

A profile picture to show that these are really 3-d:

Solid Shapes and Their Nets (I think Jo Edkins is the author) has a nice discussion of nets and a little puzzle game to distinguish nets that fold into the platonic solids and which don't. Feel free to try to do this for the icosahedron or dodecahedron!

A two dimensional challenge?

J2 asked me about a triangle with two right angles. Of course, we all know that a triangle can't have two 90 degree angles, right? Well, this fits in my list of math lies from this old post: 23 isn't prime.

We talked briefly about triangles on a plane and agreed that two 90 degree angles wouldn't work. If we try by starting with a side and building two right angles on that side, we just get parallel lines. Ok, that's standard.

However, what about this:

We discussed spherical geometry for a bit. Not shown is our first attempt on the other side of the tennis ball that did have two right angles, but the third wasn't. He wanted to see an equiangular triangle on a sphere. This led to further discussion of life on a sphere:

  • what is the largest interior angle sum for a triangle?
  • what is the smallest interior angle sum?
  • are there any parallel lines?
  • are there any squares? are there even any rectangles?
  • is π (the ratio of the circumference to diameter of a circle) a constant?
  • do we have to think like this in real life because the earth is close to a sphere?
Picking up the point about π, it occurs to me that this is another math lie. This point got a nice treatment recently by SMBC:

I made this pic small so you will go to the site and look at Zach Weinersmith's other awesome work
Some open follow-ups:
  • what is an equiangular quadrilateral on the sphere?
  • what about hyperbolic geometry? I think no triangles with two right angles. If I recall correctly, triangles there have angle sums smaller than 180 degrees (or π radians, ha!)

Other Misc Math

Now for my usual grab bag of other things we've been doing. Some of these are really great activities, so don't skimp on this section!

Desmos Function Carnival
We learned about the Desmos Function Carnival activity from a post by Kent Haines which, in turn, we learned about via Michael Pershan's post. They both were writing about teaching functions to their students and have outlined a really nice sequencing of lessons, if you're into that kind of thing.

In case you are, you might like to know that there is another flavor of Function Carnival available through the Desmos teacher site with 2 other activities. Each version is worth checking out because both of worthwhile activities that aren't in the other. Also, Kent links to this nice graphing activity from the Shell Center which also worked well with J1 and J2.

For our purposes, I was more interested in the mathematical physics (plots of position vs time or velocity vs time) and the fun of creating wacky animations with impossible plots.

If I can figure out how to post the animations, I will update this page, since the animations really enhance the experience. For now, let me give you some screen caps of some of their proposed graphs from the Cannon Man (height) activity:

Have a free-form drawing tool? Make your name!

Note: the vertical stripe doesn't really work with this animation, but the patriotic spirit is there!

Our first attempt at Quantum Field Theory

Here is J2 playing with the function carnival, but J3 (4 years old) enjoyed it just as much:

Different representations
J3 is working on place value and playing with different representations of numbers. Here, she's got the 100 board, an abacus, and foam decimal models. Sometimes I challenge her to make a number, sometimes she challenges me.

Tuesday, February 21, 2017

RSM International Math Contest

Sometime the first week of February, the RSM offers an on-line math competition. For the second year, I had J1 and J2 work through the problems at grade 3 and grade 4 level.  This post is about our thoughts on competition, but I end with a problem from the contest that had us debating.

Compete and win!

To get it out of the way, let me explain my interest in the competition and why I had the kids enter. Above all, I was curious about the problems and expected they would be interesting challenges. We found the puzzles last year interesting, so I was pretty sure this year's collection would also be nice.
While I know that there are tons of excellent activities, I couldn't resist making use of this resource when someone thoughtful had put together a convenient collection in one place.

Second, I am curious about levels and assessments. What types of questions does RSM think 3rd and 4th graders should be able to answer, but find challenging? How difficult would our two little ones find them? Since we are currently operating outside a standard curriculum framework, it is hard for me to judge where they are or what we should expect of them. Admittedly, there are other ways I could form this assessment, but they take more effort.

Notice that I don't really care about how well they perform on the test. It is interesting information for me, but I don't need them to do well.

P has a different view. She was an olympiad kid and sees competitions as the easiest way for our kids to distinguish themselves. You know the anxiety: if they don't win competitions, they won't get into Harvard, they'll end up on the street somewhere.

Perhaps unsurprisingly, the kids are getting a mixed message about the importance and reason for participating in competitions. This year, J1 was particularly sensitive. He was very resistant to doing the RSM test. I spent a lot of time talking with him. Mainly, I wasn't hoping to convince him to do the test, but I wanted to explore other issues related to the expectations he feels on himself, his relationship with J2, and his mindset about his own learning.

I am trying to communicate:

  1. effort and progress are important, starting point and base ability aren't. I know this is debatable, but I'm talking about the differences between my kids where I'm on pretty firm ground.
  2. there are many aspects to math and mathematical ability. Calculating and answering questions quickly are facets, logical reasoning, strategic thinking, spatial reasoning, asking good questions, gathering data, exploring connections, etc, etc are all components, too.
  3. time isn't important. One issue with the RSM test is that it is timed. This goes in the face of our repeated efforts to emphasize that the time it takes to solve something is not a key consideration. To help with this, I screen capped the questions so he could work on them after the official time had expired. Note, however, his recorded performance on the test was based on work within the contest rules. 
  4. Math is not about right answers.
The next section might help illustrate that last point.

Which area do we want?

As mentioned above, one of the questions from grade 4 cause us to argue among ourselves. I'm paraphrasing slightly:
Two tennis ball machines stand on opposite ends of a 25 meter by 10 meter court. The yellow machine shoots yellow balls that stop on the court 2 meters to 16 meters from the yellow machine's side. The green machine shoots green balls that stop on the court 5 meters to 20 meters from the green machine's side. Find the area of the court that has balls of either color on it.
Here are some ideas from our discussion:

  • Option A: we want to find the area of overlap, because that's the only place we could find balls of either color.
  • Option B: we want to find the combined areas, because that's where we could find tennis balls, either color.
  • Option 1: the area with green balls is a rectangle, the area with yellow balls is a rectangle.
  • Option 2: think of the ball machines as single points, shooting balls at various angles and various distances. the target area for each machine is an intersection of of a circular ring and a rectangle. This idea came from J2 when we were looking over the grade 4 questions after time had expired.
To answer the question, choose either A xor B and choose either 1 xor 2. So, what is the right answer?

I think the right answer is to have this discussion, to encourage multiple interpretations (with justification), to see how the answers compare, to think about what other things we could do to make the intended interpretation more clear (a diagram seems the most obvious), to recognize that this is part of the richness of math and it can't be represented by a single numerical answer on a timed test.

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):


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

Thursday, February 16, 2017

More Man Who Counted (gaps and notes)

As previously mentioned, we have been reading The Man Who Counted. While the story is good and there are nice math puzzles, we've found some of our best conversations have come from errors or weaknesses in the book. Here are three examples:

How old was Diophantus?

In chapter 24, we encounter a puzzle to figure out how old Diophantus was when he died. In summary, the clues are:

  1. he was a child for 1/6 of his life
  2. he was an adolescent for 1/12 of his life. (J1: "what's that?" J0: "a teenager")
  3. childless marriage for 1/7 of his life
  4. Five more years passed, then had a child
  5. The child got to half its father's age, then died.
  6. Diophantus lived for four more years
Perhaps we are wrong about our interpretation of the clues, but we noticed two things:
(a) the answer is not a whole number of years.
(b) the answer given in the book doesn't fit the clues.

For the first part, it seems a natural assumption of these types of puzzles that we are only working with whole number years. Sometimes, this is an interesting assumption to directly challenge.
Here, since the clues involve a second person (Diophantus's child) we felt whole numbers were a strong assumption. Also, the name Diophantus, you know?

Each clue required some discussion for us to agree on the interpretation. The one that seems most open is the fifth clue. In particular, did the child live until its age was half of the age of its father at the time of birth or to the point that, contemporaneously, it was half its father's age?

For completeness, I'd note that neither interpretation matches the book's answer. The first interpretation does allow a whole number answer, but it doesn't give whole numbers for all the listed segments of Diophantus's life.

Just so you can check for yourself, the solution given in the book is 84 years old.

How do you fix it?
We discussed several possible fixes:

  • accept answers that aren't whole numbers or require whole number segments for each clue. This allows us to take the alternative interpretation of the fifth clue (though that still isn't satisfying) or to accept the clues and just take a new answer. This isn't satisfactory because... Diophantus.
  • Change clue 4 or clue 5 to match the book's answer. This approach seemed to fix the puzzle without distorting it or changing the mathematics required to analyze it.
  • Change clue 1, 2, or 3. While possible, these seemed to open the possibility of changing the character of the puzzle. Also, these fractions were plausible based on our own experience of human life spans.
Of course, an even more satisfying answer would be to introduce a further variable and make the puzzle into one that makes heavy(ier) use of the integer restriction.

Clever Suitors

In chapter 31, Beremiz is confronted by a nice logic puzzle. Three suitors are put to a test, each is blindfolded and has disc strapped to his back. The background of the discs: other than color, the discs are all identical, there are five to choose from, 2 black and 3 white.

The first suitor is allowed to see the colors of the discs on the backs of his two competitors, then required to identify the color of his own disc and explain his reasoning. He fails and is dismissed.

The second suitor is allowed to see the disc on the back of the third suitor, then required to identify the color of his own disc and explain his reasoning. He fails and is dismissed.

Finally, the third suitor is required to identify the color of his own disc and explain his reasoning.  He succeeds.

Weakness 1
As a logic puzzle, we enjoyed this. Our problems came from the context in the story. This challenge was set to the three suitors as a way of fairly judging between them by finding the most clever suitor. However, this process was clearly unfair. In fact, it is inherent in the solution that it was impossible for the first and second suitors to determine the color of their own discs.

This led to a nice discussion about who really held the power in this process: the person who structured the problem by deciding what color disc should be on which suitor and what order they would be allowed to give their answers.

  • consider all arrangements of discs. Are there any arrangements where none of the suitors can answer correctly?
  • What is the winning fraction for each suitor? If you were a suitor, would you prefer to answer first, second, or third?

Weakness 2
Our second objection was non-mathematical, but again related to the story context. The fundamental problem wasn't how to choose a suitor. The fundamental problem was how the king could remain peacefully friendly toward all the suitors' home nations through this process.

For this discussion, we went back to the story of Helen of Sparta, which we'd read a long time ago in the D'Aulaire's Book of Greek Myths. Of course, that also led to discussion of the division of the golden apples, another puzzle we all felt surely could have been solved more effectively with some mathematical reasoning...

The Last Matter of Love

The last puzzle of the book is in chapter 33. It is another logic puzzle, again intended to test the merit of a suitor in marriage. The test:

  • there are five people
  • two have black eyes and always tell the truth
  • three have blue eyes and always lie
  • the suitor is permitted to ask three of them, in turn, a "simple" question each.
  • the suitor must determine the eye color of all five people
As a logic puzzle, we readers get some extra information:
  1. The first person is asked: "what are the color of your eyes?" The answer is unintelligible.
  2. The second is asked: "What did the first person say?" The answer is "blue eyes."
  3. The third is asked: "What are the eye colors of the first and second people?" The answers are "the first has black eyes and the second has blue eyes."
Simple questions
Our first objection was the part about asking "simple" questions. Having developed our taste for these types of puzzles through the knights and knaves examples of Raymond Smullyan (RIP, we loved your work!!!), the third question really bothered us. If you're going to go that far, why not ask the third person for the color eyes of all five people?

Personally, I would prefer that the puzzle require us to ask each person a single yes/no question.

As an extension: can you solve the puzzle with that restriction? 

Getting lucky
Again, we felt that this puzzle didn't meet the requirements of the context: to prove the worthiness of the suitor. Putting aside the question of whether this is really an appropriate way to decide whether two people should be allowed to marry, the hero here got lucky.

Extension: what eye color for the third person would have caused the suitor to fail?
Extension: what answer from the third person would have caused the suitor to fail?
Extension: for what arrangement of eye colors would the questions asked by the suitor guarantee success?
Extension: what was the suitors' probability of success, given those were the three questions asked?

Our final objection was the simple waste in the first question. From a narrative perspective, this is justified and even seems made to serve the purposes of the suitor. However, it opens another idea:
can you solve the puzzle, regardless of eye color arrangement, with only two questions?

Feel free to test this with yes/no questions only or your own suitable definition of a "simple" question.

The power of...

As a final thought, let me say that I think errors and ambiguity in a text are a feature, not a bug. It is another great opportunity for us to emphasize that mathematics is about the power of reasoning, not the power of authority.

Tuesday, February 14, 2017

Good games and bad

Recently, we have been playing the following games:

  1. Go (baduk, weiqi, หมากล้อม). For now, we are playing on small boards, usually 5x5 or smaller.
  2. Hanabi
  3. Cribbage
  4. Qwirkle (not regulation play, a form of War invented by J3 and grandma)
  5. Munchkin
  6. UNO
  7. Vanguard
I've ordered these by my own preference. In fact, I would be delighted playing just the first two exclusively and am happy to play cribbage or Qwirkle when asked.

For the other three, I find myself biting my tongue a bit and grudgingly agreeing to be part of the game. I'm in the mood for strategic depth and a moderate (but not large) amount of pure chance. Part of my feeling was echoed in a recent My Little Poppies post: Gateway Games.

However, as in the MLP post, I recognize that my enjoyment of the game is only a part of the reason for the activity. I guess the kids' enjoyment counts, too. 

Beyond that, even the games with limited depth are helping to build habits and skills:
  • executive control: assessing the situation, understanding what behavior is appropriate, understanding options and making choices.
  • general gaming etiquette: taking turns, use of the game materials
  • meta-gaming: helping and encouraging each other, making sure that the littler ones have fun, too
  • numeracy and literacy: every time a number or calculation comes up or when something needs to be read, they are reinforcing their observation that math and reading are all around them.
  • meta-meta gaming: game choice, consensus building, finding options that interest and are suitable for all the players, knowing when it is time to play and when it isn't.
As a family, and a little team, they are also building a shared set of experiences and jargon as they absorb ideas from each of the games.

All of these are, of course, enough reason to make the effort to be open minded and follow their gaming lead.

Monday, February 13, 2017

NRICH 5 Steps to 50

A quick note about the game we played in first grade today: 5 Steps to 50.

This is an NRICH activity that I've had on my radar for a while. I even made a pencilcode program to explore the activity in reverse. True to their other activities (check them out!!!) 5 steps to 50 requires very little explanation, is accessible to students with limited background, but has depth and richness.

Our lesson outline
I explained the basic activity and did an example at the board. To get my starting value, I had one student roll for the 10s digit and one for the 1s digit. Then we talked through together as we added 10s and 1s.

I then distributed dice and had the kids try 3 rounds. As they worked, I confirmed several rules:

  1. the only operations allowed are +1, -1, +10, -10
  2. we must use exactly five steps (I note that this is ambiguous on the NRICH description, they say "you can then make 5 jumps")
  3. we are allowed to do the operations in any order
  4. we can mix addition and subtraction operations
After everyone had been through 3 rounds, we regrouped to summarize our findings:

  • Which starting numbers can jump to 50?
  • Which starting numbers cannot jump to 50?
We helped the kids resolve disagreements and then posed the following:
  • What is the smallest number that can jump to 50?
  • What is the largest number that can jump to 50?
For those to challenges, we kept the restriction that the numbers must be possible to generate from 2d6.

Basic level
To engage with the activity, some of the kids just started trying operations without much planning. This quickly reinforced the basic points about addition and place-value and commutativity of addition.

For these kids, it was helpful to ask a couple of prompting questions:

  • What do you notice? This is a standard that never gets old!
  • If this path doesn't get to 50, does that mean there is no path to 50?
This second question, particularly, raises the interesting observation that it is easy to show when a number can jump to 50 (just show a path) but to show that no path is possible requires a different type of thinking.

Getting more advanced
The next level of sophistication was really about noticing that the key consideration is the distance to 50. In particular, this identified a symmetry, where n could jump to 50 if 2*50 - n can jump to 50. Of course, the kids didn't phrase this relationship in this way....

The next major step is thinking about a way to systematically write down the paths.

Wednesday, February 1, 2017

Perfect Play for My closest neighbor

Joe Schwartz at Exit10a wrote a fraction comparison post that prompted me to write up more of my experience and thoughts on this game.

Let's find perfect play
This week, I intended to use the game one last time with the 4th graders as an extended warm-up to our class. The challenge I presented:

If we got super lucky and were given perfect cards for each round of the game, what are the best possible plays?

My intention was to spend about 20 minutes on this. Depending on how quickly it went and the kids' reactions, I considered giving them a follow-up for a short homework: what are the best plays if we include all cards A (1) through K (13)?

How did it go?
In the end, the basic activity took the whole class. These comparisons were difficult for the kids, so we spent time talking about each different strategy for comparison:

  1. common denominators
  2. common numerators
  3. distance to 1
  4. relationship to another benchmark number. Like 1/2 in Joe's 4/6 and 8/18 example, a benchmark is a "familiar friend" that should be relatively easy to see it is larger than one and smaller than another. In practice, 1/2 seems to be the most popular benchmark. 

For visualization, drawing on a number line seemed to work best.

I did not assign the full deck challenge as homework. Instead, we gave them some more work with fractions of pies and bars.

What have I learned?
This game is really effective at distinguishing levels of understanding:
(0) some kids are totally at sea. They don't really understand what this a/b thing means, how a and b are related, etc. These kids struggle with the first round of the game when the target is 0, when the idea is to just want to make their fraction as small as possible.

(1) Some kids have got a basic understanding of the meaning of the fraction and can play confidently when the target is 0 or 1. They might still be weak about equivalent fractions. Trying to play some spot-on equivalents when 1/3 and 1/2 are targets is a give-away.

(2) familiar with some frequent friends: kids who can tell readily whether their plays are larger or smaller than the target for 1/3, 1/2, 3/4.

(3) proficient: have at least one consistent strategy they can work through to make a comparison

(4) fraction black-belts: using multiple strategies, already familiar with many of the most common comparisons.

What would I do differently?
Generally, I think it is valuable to spend more time and more models directed at the basic understanding of what fractions mean. The kids who were at or close to stage 4 have, over the years, been seeing diagrams of pies, cakes, chocolate bars, number lines and physical experience with baking measures and fractional inches on measuring tapes and rulers. Oh, and also actual pies (mostly pizza), cakes, cookies, and chocolate bars discussed using fractional language.

More locally, for this game in a class of mixed levels, I would

  • lean toward doing this more as a cooperative puzzle
  • re-order the targets for the rounds as 0, 1, 1/2, 3/4, 1/3, 2 (note: I don't have strong feelings about where 2 fits in this sequence)
  • I also would consider allowing equivalent fractions to the target as winning plays

Impassable Din Daeng (BKK intersections 3)

It has been a while since I've done one of these.  For your topological and civil engineering pleasure, I present  Din Daeng intersection (แยกดินแดง):

Notice how the expressway obscures key details?

Magnified view. We now see a tunnel, but where does it emerge?

This intersection has a special place in my heart. Last month, J1 and J2 played in a squash tournament at the Thai-Japan Youth Centre. It isn't visible on either map I've included, but is just to the northeast of the intersection.  Since we live on the west of the intersection, we needed to cross somehow.

After three failed attempts (following the driving directions on google maps) to get through from west to east, I ended up parking our car in one of the small side streets and we just walked. The walk took about 30 minutes...

ideas for upcoming classes

warm-ups for all

WODB: (1) shapes book (2)
any: Traffic lights/inverse tic tac toe/faces game
good options here, mostly grades 1/2: some games
dots & boxes (maybe with an arithmetic component)

Grades 1 and 2

close to 100 game: 
Equipment: A pack of cards with 10 and face cards (J,Q,K) removed.
- Deal out 6 cards to each player
- Each player picks 4 cards from the 6 cards they were dealt to form a pair of 2-digit numbers.  The goal is to get the sum of the two numbers as close to 100 as possible but cannot exceed 100.

Grades 3 and 4

Factor finding game (maybe warm-up?)
Factors and Multiples game
Contig for 3 and 4 (explanation).
Times tic-tac-toe: review for Grades 3 and 4
Fraction war for grade 4 (smallest card is numerator)
Multiplicaton models: worth making for grades 3-4 for solidifying concepts? Associated games
d 2

card on head game

Pico Fermi Bagel

Magic triangle puzzles

damult dice

(1) Dice game perudo
- multi-player, 2-5
- Everyone gets the same number of 6 sided dice (full game they get 5, I would start with 3)
- Everyone has a cup to shake and conceal their dice

Basic Play
- Simultaneously, players shake their cups and turn them over on the ground or a table. They peak in to look at their own dice, but keep them concealed from the other players.
- starting randomly (or from the person who lost a dice in the last round), players make bids, for example: two 3s. 
This bid signifies that the player has 2 (or more dice) showing the value 3.
- the next player has two choices: 
  • call/doubt the previous player's bid: if they do this, all players show their dice. If there are enough to meet the bid, the caller loses a die. If not, then the bidder loses a die.
  • raise, either the number of dice or the value or both get increased 
Advanced rules
- Ones are wild, they count as any number toward the target bid
- If someone drops to their last dice, they start the next round. On that round, only the number of dice can be increased in the bid, not the value. Ones are not wild on this round
- After someone bids, the next player has a third option, to call "exact." If the bid is exactly matched by the dice, then the bidder loses a die and the caller gets an extra one. If the actual dice show either more or less than the bid, the caller loses a die.

remainder jump
we played this game before, but we could give them blank boards and let them create. See the last page here:

(1) double digit and double dollar:
We've done something like this, but I think there could be a good variation done trying to make 1000 baht, using 1, 2, 5, 10, 20, 50, and 100 baht units.

(2) biggest rectangle. This could be used as a warm-up. For the older kids, they will probably have seen something like this, but I like the inclusion of perimeters that are even but not divisible by 4 and odd perimeters and the question about "smallest area" (here are 5 questions).

(3) some of these games are promising:

Tuesday, January 31, 2017

Quadratic Friends (The Man Who Counted)

J1, J2, and I are currently reading The Man Who Counted. Here are some quick thoughts:

Quadratic Friends
The book is a great entry point for mathematical discussions. In fact, it makes it questionable as bedtime reading, since I have to be careful to find a more narrative section to close the evening. Otherwise, we would just continue talking and they'd never get to sleep.

Fortunately, the J's are willing to extend some of these conversations over to the next day, so we're not obligated to wrap up everything in one evening.

Here is an example discussion: in one of the early chapters, the protagonist Beremiz talks about the special relationship between 13 and 16. Namely:
13 * 13 = 169
1 + 6 + 9 = 16
16 * 16 = 256
2 + 5 + 6 = 13
Finding more
We wondered: what other pairs of numbers share this property?

Our first instinct was to gather data, so we started calculating some examples. We began with 0 and worked up, squaring, adding the digits, repeating. We found a couple of cases that flowed into the 13-16 relationship, for example 7. This gives a feeling that 7 is very fond of 13, but 13 only has eyes for 16.  Not the usual way people think about numbers, I guess.

Along the way, we made some interesting observations about this iterative process. I won't spoil the surprise, but would encourage you to explore yourself.

I'd note that J1 did the calculations up to 30 in his head, while I was a bit lazy and wrote a pencilcode program.

An extension
This conversation branched in an interesting way. Squaring is a natural thing to do with numbers, but summing the digits is a bit artificial. It depends on a choice of base. So, a natural follow-up question:
what quadratic friends exist in other bases?  This is an exploration for another day.

Tuesday, January 24, 2017

Delta Pack the uncreative way (euclidea series)

Returning with a very brief installment of geometric constructions. Basically, I was just brute-forcing the constructions in this pack, so most of the cases where I managed the minimum move constructions were just because those were straightforward.  Reviewing the pack in preparation for this post, I did find a couple of ways to shave down some of the constructions, but I don't feel like there was a major breakthrough.

Here's a snapshot showing (almost) where the V stars are:

If you do some simple counting, you'll see that there is one more V-star missing. Since this somehow escaped me the first time, I'll leave its location as an exercise for you, too.

I've mentioned the idea from 4.5 before and I still like that construction, even though it is very simple.

Constructing the two equilateral triangles also seemed nice. Constructing the inscribed and circumscribing circles from the triangle seems more like the "usual" direction of construction, so these reverse constructions appealed to me.

Monday, January 23, 2017

Fractions and Farey Addition

Benjamin Leis (who posts at Running a Math Club) flagged this video in response to our recent fractions work: Funny Fractions and Ford Circles (Numberphile). 

Ex ante discussion ideas
The video gave me several ideas for possibly interesting conversations with the kids:
(1) Some basic geometry, particularly for J3. Circles that are tangent, nesting pictures, pictures that have fractal qualities.
(2) Comparing Farey addition and regular addition
(3) Well-defined operations on fractions. I always like to discuss whether the operations gives us the same results regardless of the equivalent form we start with? Farey addition is a good example where the choice of representation is important (indeed, Prof Banahon is careful to keep reminding us that he wants the fractions in lowest terms.)
(4) why do we want the fractions in simplest terms? Possibly relate this to the Cat in Numberland (showing rationals are countable).
(5) what happens if we try Farey addition of three fractions in a row: e.g., (1/5) @ (1/3) @ (1/2)? This is one of the few "naturally occurring" non-associative operations I know.
(6) Since associativity doesn't work, surely distribution of multiplication over Farey addition must not work, right? What about commutativity?
(7) Linking back with our comparison game, if a<b, how do a@c and b@c compare? If a@c < b@c, what can we say about a and b?
(8) what if we allow negative numbers? How should we define Farey addition, then?

How the conversation actually went
J2 was especially taken with the picture of the Ford circles and immediately had two requests: he wanted to draw them and he wanted me to create a pencilcode program to draw them.

The former was a great activity with a lot of figuring and fraction practice. Here he is, hard at work:

Along the way, there was lots of discussion about where to position each fraction on the number line (he scaled with 20 cm as the unit distance from 0 to 1), and how big to make each circle. Tangency condition was a nice check on his work. He would see right away when something was wrong (which did happen several times:

We did talk through some of the ideas on my pre-planned list: is Farey addition well-defined on fractions (no! point 3), does associativity work (no! point 5), could we extend to negative numbers (yes, make the numerator negative seems to work best, point 8).  Other areas are still open for future discussion.

Pencilcode result
I wrote a quick program here: FareyFord. You'll notice that it doesn't actually generate Farey sequences. Instead, it creates generations of fractions, starting from 0 and 1 as the original parents. For each new generation, it uses Farey addition to create a new fraction between each adjacent pair in the previous generation.

Here's a picture of the associated Ford circles:

This method raised an interesting question: what is the largest denominator in each generation? If you don't know, it is cute and worth considering.

More Go (miscellaneous)
Note: this part is unrelated to fractions or farey sequences.

J3 wasn't in the mood to play more capture go with me, but I had an idea. I noticed in one of Nick Sibicky's lectures that one of his students was a young girl, roughly around the age of our three kids. I showed that part of the video to J3 and she made the connection: "this is something girls like me do."

We went and played some silly games on very small boards: 1x1, 2x2, 3x3. In the picture below, we set out a blue-green alternating boundary around a 3x3 board. Then, I asked J3 how many different moves were available. She pointed first to the center, then I asked if there were any other spaces that were the same as the center, if we moved the board around or tipped ourselves upside down.

No, so we made the center red. What other moves? She then chose a side square and figured out that there were three other places that were equivalent. Those became yellow. Finally, we figured out that the four corners were also identical, so that gave us the final picture:

Later, I was playing 9x9 with J2. Instead of go stones, we used Banangram tiles for the white stones. At the end of the game, we tried to make words with the captured tiles from the game. Here was one case where we could (sort of?) make a complete scrabble chain with all the captures:

Wednesday, January 18, 2017

Closest neighbor one-on-one

In my last post, I wrote about playing Denise Gaskins' closest neighbor fraction game with our 4th grade class. Yesterday, I spent time with J2 and used the game as a semi-cooperative puzzle.

This activity worked really well and the experience gave me some additional ideas about how to use the core ideas again with the 4th grade class.

Puzzle or game?
First, there were only two of us, one a kid and another an adult, so that background naturally makes the activity very different. As the key modification for play, we played all of our hands open and helped each other find the fraction in each of our hands that was closest to the target for that round. Then, we worked together to determine which of those two "champions" was closest overall.

Some of the consequences:
  • the activity was not really competitive (see below)
  • J2 had to do a lot more fraction work.
Let me explain the second point here. Because we were looking for the best play, J2 had to consider all of the combinations in his hand (20 choices). Some of those can be rejected quickly with simple analytical strategies depending on the target. Even this is good number sense thinking. Also, some combinations are close competitors and need to be analyzed more carefully.

If we were playing with closed hands, he could choose two cards, play a fraction based on them, and I wouldn't be able to say anything about whether those were his best options or not.

Second, while I write that "we worked together," as a sneaky dad, that means that I pretended to do work, while actually getting J2 to analyze my hand as well as his. Really, the only thing I offered was an alternative comparison strategy, once he had already worked through his own approach.

An example of some strategies
We found that some of the comparisons that arise naturally in this game are quite tricky, even for me. For example, quickly tell me which is closer to 1/3: 1/5 or 4/9?

We found that placing the fractions on a number line was a really helpful strategy for many of the comparisons. We also made very heavy use of the two strategies involving common numerators or common denominators.

Finally, you can see in this example that J2 is comfortable mixing decimals and fractions, for example converting to 1/2 to 3.5/7 to aid some comparison:

Our grid
Through our play, we filled out this grid, taking turns putting in our best results and congratulating each other when our hand was the ultimate champion for that round:

Competition and Strategic thinking
I was particularly pleased by one comment J2 made about this overall game: "this is mostly luck, how well we can play depends on the cards we get." This comment came after one round where he had several duplicate cards in his hand, reducing the number of distinct values he could play. We've discussed elsewhere my goals of helping the kids think about game structure, so I always love it when they bring those ideas up themselves.

Some thoughts about competition. While we played this game non-competitively, I'm not opposed to competition nor do I think that this game always needs to be played non-competitively. Ultimately, my litmus test is how to play in a way that is the most fun. If I were a more serious educator, I suppose I would also consider which way is the most educational, too.

It won't always be obvious what is the best way to play each game. In this case, I got to benefit from the prior experience with the class and my close knowledge of J2. Many times, I'll tell the kids that there are several ways to play and we'll try them out together, then review the experience.

Among other things, this is why I love handicap games like Go. By adjusting the starting advantages, we can create scenarios where it is very competitive and very fun, even though the players have very different levels of experience and current strength in the game. And also, there are things we can do together when we want a non-competitive activity.

Ideas for going back to class
From this time with J2, here are my ideas about taking the game back to the 4th grade class are:
  1. Spend a lot of time on fraction comparison strategies before we play
  2. Reduce the number of cards dealt to each player
  3. play as teams
  4. convert to open hands with a lot of talk about why we chose particular plays

An actual puzzle

As a reward for reading down this far, here's an actual puzzle related to the closest neighbors fraction game:

During the round where the target is 1/2, Jay plays 6/6 = 1. Was that her best play? How do we know?