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:
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.
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.
Favorites
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.
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:
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:
Spend a lot of time on fraction comparison strategies before we play
Reduce the number of cards dealt to each player
play as teams
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?
Remember, we are playing our game with a single deck of playing cards and each player is dealt five cards
The target for the next round will be 3/4
Burning 2 identical values might allow Jay to increase the diversity of her hand, especially if she happened to have 3 or 4 sixes.
Hey, life doesn't promise that all puzzles will have solutions that can be wrapped up in a nice neat package, does it?
Denise Gaskins recently flagged a post about a good fraction game: My Closest Neighbor. I tried this out in class today.
A pre-test
First, I wasn't sure whether the level of the game would be right for the kids. I was considering it for the 3rd and 4th graders, but had some alternative activities planned in case. To start, I posed the following questions:
Which is closest to one-half: 1/3 or 2/5? The third graders really struggled with this, so I left it alone and went to my plan B games. The fourth graders were all confident on this one.
Which is closest to 3/4: 5/11 or 11/12? This was a challenge for the fourth graders, but I thought it would be ok to play the game.
In our discussion of the second question, we explored two strategies:
making a common denominator
comparing with reference numbers
The common denominator is a bit of a pain, since 11 is prime, though at least we have the fact that 4 is a factor of 12. One student soldiered through this approach, but it was difficult for the other kids to follow.
For the second strategy, we made use of some observations that were more elementary for the kids:
(a) 5/11 < 5/10 = 1/2
(b) 3/4 is halfway between 1/2 and 1
(c) 11/12 < 1
Combining these, it was easy for us to draw a rough number line, place 5/11, 1/2, 3/4, 11/12, 1 and see that 11/12 must be closer.
The game
We played three rounds: target 0, target 1/3 and target 1/2. I think this game was very challenging for the kids. Everyone had to work to figure out the best play from their hand and didn't always make the right (local) choice. For example, whether to choose 5/8 or 5/9 for a 1/2 target.
Once everyone had played, the challenge was still just starting. They had to figure out who was closest. I structured the discussion by helping them figure out which plays were lower than the target and which were higher. For the ones that were lower, they could put them in order and only needed to consider the highest. Then, we worked on the ones higher than the target and got the lowest of those.
In the course of this discussion, we added a third strategy to the ones listed above:
making a common numerator
Summary thoughts
Fraction comparison like this was still too difficult for the kids to make an engaging game. If I were to do it again, I would change to make it more of a puzzling exercise, removing competition and any sense of time pressure.
Once the kids gain a bit more experience, though, I think this game has some nice features. It is particularly good for practicing fraction sense, and the multiple rounds allow some scope for strategic play.
In past posts, we've shown some of the make-shift materials we are using to play/learn Go without a proper set. Over the last two days, we had experiences that reinforce the value of this approach.
Exploring an earlier pattern
First, when playing with J3, she noticed that we could complete a repeating blue-green-yellow-red pattern around the boundary of a 5x5 board. In a follow-up conversation with J1 and J2, we explored this:
J1 explained why it would work, grouping the boundary tiles as 5 for each side of the playing square and 4 in the corners, so 5 x 4 + 4. This made it easy to see that the boundary would be a multiple of 4 and also made it easy to extend to any square board: n x 4 + 4.
J2 had a new idea. He thought we were talking about the pattern continuing as an inward spiral. That gave us this design:
This also led to discussions about symmetry (the blue and yellow have reflectional symmetries that green and red lack) and further investigation on boards of different sizes. Interestingly, we found that, for some boards, none of the colors have a reflection symmetry.
Trying some tsumego
I set J1 and J2 the following challenge (J3 was watching): are the blue cubes alive or dead in each of these two clusters?
Putting aside the interesting Go discussion that resulted, there are two consequences of doing this on the 100 board: extra unnecessary information and a built-in coordinate system. By unnecessary information, I'm talking about the letters on the white tiles and the numbers on the 100 board. This is information that is entirely orthogonal to solving the life-and-death puzzles. This is a simple toy version of one of the key modern challenges in problem solving: identifying which information is useful and which is a distraction.
On the other hand, for talking about the puzzles, we could say things like "what if white plays a tile on 99?" For J3 who was watching, this offered another little example of the idea that numbers are all around.
Some capture fun
For the last example, I set out some white tiles (some alone, some in groups) and asked J3 to capture them with blue cubes. After we did that, we counted the number of cubes we used to capture by moving them to cells in the 100 board (note that we removed one of the lone white tiles). A fun counting exercise, an opportunity to talk about groups of 10, and more familiarization with the layout of the 100 board.
Since I've branched into this topic, I want to include some notes on additional references I've found and some results I've been able to get. First, an admission:
I'm still having trouble finding the intersection points of a circle and a line when the circle's center is on the line!
Another resource
James King (UW home page) has a nice session outline working through compass only constructions. Be warned, there are some spoilers for some of the Euclidea challenges in that material. Unfortunately, I can't tell from write-up when and where this was used. Maybe this NWMI meeting?
In Professor King's notes, he describes the construction I'm struggling with as "important and difficult," so I will have to redouble my efforts.
Non-collapsing compass from collapsing compass
As an aside, perhaps you have noticed that everyone else refers to the two types of compasses as collapsible and non-collapsible. Doesn't that terminology strike you as wrong, too? The point isn't that one type of compass is able to collapse, but rather that it always collapses when not drawing a given circle. Also, a common form of "non-collapsible" compass is able to collapse!
Just squeeze the legs together to collapse
Anyway, I was able to figure out the construction of a non-collapsing compass, just in time for Euclidea 13.2.
I'll give a construction below. For a more mild hint, the key is the ability to reflect points through a line we've already "constructed" (where we have two points on the line).
I've chosen to do this in GeoGebra so I can show a sequence, explain some of the reasoning, and grey out earlier parts of the construction that we don't need anymore. Check my work to make sure I didn't slip in any subtle straightedge moves!
Here, we are given points A, B, and C and want to construct the circle centered at B with radius AC.
First step is two circles centered at A and B with radii AB, giving us points D and E. The line DE will be the one we are reflecting through. Key is that A and B are the reflected images of each other through this line.
Next, we draw the circle at A with radius AC. This intersects our circle centered at B in point F. F is a special point, distance AC from A and AB from B, and distance DF from D.
That means the reflection of F through line DE is a point distance AC from B! We can find it using the fact that it will also be distance AB from A and distance DF from D:
It almost looks like C and F are the same distance from D. That isn't true and is entirely coincidental. Try it out yourself and use the move tool to push C around and see that this is the case. In particular, you might notice that we could have made another choice for F (it is a point of intersection of two circles, we could have taken the other) and that other choice is on the opposite side of AB from C.
I'll leave to you the satisfaction of completing the construction and drawing our target circle.
Also, the Euclidea challenge is formulated very slightly differently: instead of point C, we are given the starting circle centered at A. That means the construction is one move shorter than what I've shown here.
Recently, I have been insinuating Go playing into my time with the 3 Js. This was initially motivated by a quote I saw on one of the Mathpickle pages (Gamers under Inspired People):
Schools should experiment teaching go* instead of a regular math curriculum for one year to students around the age of 7. It is my prediction that the strong problem solving skills that this will engender will make superior students than any existing mathematics curriculum.
Now, when we first decided to have kids, my objective was to help them develop into people with whom I would enjoy spending time. In particular, I wanted to be able to play games with them. With that in mind, the Mathpickle idea resonated with another idea from Richard Garfield (via Math Hombre):
play each game so as to increase your chances of winning all games
With these three ideas in mind, I went looking for a way to properly introduce Go to our clan.
Curriculum outline
Not surprisingly this is a question other gaming and math people have asked before. Quickly putting together the ideas I liked the most from other sources, we basically started following the curriculum shown in the Go GO Igo videos with Yoshihara Yukari (Umezawa Yukari at the time of filming):
Since we don't actually have a Go board or stones, we started with the electronic board CGoban. This works well for J1 and J2. We have also used J1's chess/checkers set as a makeshift 9x9 board (playing on the lines instead of the squares).
For J3, we started playing the simple capture game using the blank side of our 100 board.
For the first lesson, we arranged things like this:
She played the centimeter cubes (which substitute for black stones) and I played the Bananagram tiles (substituting for white stones). I gave her a four stone advantage and we played three games with me starting in different places (center, corner, side) and saw that she could easily capture at least one of my stones without trouble.
Some of J3's observations along the way:
There are 11 blue tiles forming the boundary
There are 25 squares in our playing area
There are five squares along each edge of our playing area
The placement of the four handicap stones is symmetric in the playing area. There are several symmetries
Stones in the corner have two directions to live
Stones on the edge have three directions to live
Stones inside have four directions to live
For the second lesson, we made the board a little differently based on J3's preference for a blue-green-yellow-red pattern around the border:
This time, J3 made some different observations:
The pattern continues around the border (at no place, did we have to break the pattern). A more advanced question: will this always happen with our Blue-Green-Yellow-Red pattern around a square board?
The colors in opposite corners are the same (blue-blue and yellow-yellow)
There are more than 11 tiles on the border now.
Still 25 squares on the board and 5 squares along each side
For the third session, J3 was willing to reduce her starting advantage and she wanted to place the handicap stones herself:
This is a losing position (remember, we are still playing where the first to capture at least one stone is the winner):
She didn't take losing especially well, but this is a nice feature of playing these kinds of short games. The kids can make a mistake, they have to deal with failure, but it isn't very costly since each game only takes a couple minutes and the next game starts right away.
Some Go concepts we are still developing
At this stage, we are still working on the basic concepts:
once placed, the stones don't move
only the main compass directions (north, east, south, west) are liberties. Diagonals don't give life.
liberties are shared for a group, not just the individual stones. For example, a stone surrounded by its own color is not dead (if the overall group still has liberties).
I need to remember to announce "Atari" when a stone or group has only one remaining liberty.
Observations
From a Go/games perspective, I think it is helps to start playing a lot of low-cost games: fast games where the winning condition is easy to identify and immediate. This allows the kids to make mistakes, see clearly the consequences of those mistakes, and lose, then immediately try again.
From a math perspective, there is a huge amount of elementary math that comes out of the simple games:
counting
addition
patterns
some basic multiplication, particularly with the array model
In addition, we had the usual experience with using physical manipulatives: something extra always comes up. For example, using the 100 board inspired J3 to show off to me that she can count to 100 now (using the board as a reference).
Note: This is a little bit out of order, but reflects the challenge on which we're currently working.
Choices of Construction Rules
The classic construction problems involve specific, restricted tools:
unmarked straight-edge: this tool is only able to determine a line that contains two existing points (previously given or already constructed). It has no length markings and only one side.
collapsing compass: this tool is only able to determine a circle given the center and a point on the circle. Again, both center and an included point must already be given earlier in the construction. Like the straight-edge, the compass has no way to indicate the length of the radius. (side note: when I was in HS geometry, we didn't require the compasses to be collapsing).
A plane as a writing surface and infinitely fine/precise pens: these allow us to identify the intersection points of the constructed objects (circles and lines).
I, and maybe you, for years considered these rules to be natural. However, there are several other construction rules that are at least as natural:
Origami: moves based on folding paper. This is really a fascinating topic. Take a look at this Numberphile video for a taste.
Straight-edge only constructions: for those unfortunate to have left their compass at home.
Compass-only constructions: easy to imagine how ancients made a good compass, but how would they have gotten a really straight edge anyway?!
Several of the Euclidea challenges impose either straight-edge only or compass-only restrictions, which got me interested in this general topic again.
Examples in Euclidea
Unless I've missed some, the restriction challenges start officially in Theta pack:
Drop a perpendicular (8.4): we are only given the straight-edge, but we are also given a circle.
Mid-point (8.5): find the midpoint of a segment with the straight-edge and a parallel line.
Segment trisection (9.7): same restrictions as 8.5
Midpoint (13.1): this time, we've only got the compass, no straight-edge
Some I've not yet unlocked: Tangent to Circle (13.5), Drop a Perpendicular (13.7), Line-Circle Intersection (13.8)
In addition, Zeta pack has a some challenges that, while not marked as restrictions, are good warm-ups: Point Reflection (6.1), Line Reflection (6.2), and Translate Segment (6.6).
Humans can forget things!
If you've made it to Zeta pack, you know that the collapsing compass is able to replicate the function of a non-collapsing compass, so that particular restriction doesn't change what is theoretically constructible (but it sure reduces a lot of extra steps and auxiliary objects in real constructions.)
It turns out that the straight-edge is also unnecessary! This result is the Mohr-Mascheroni Theorem. One really fascinating/disturbing fact about this result is that it was first proven, as far as we know, in 1672 by Mohr, but his proof was lost for over 250 years. Mascheroni independently discovered and proved the result about 120 years after Mohr.
I think this is an important lesson that, with search-empowered internet, is easy to forget: human knowledge accumulation isn't always steady or certain.
Compass-only program
Cut-the-Knot (a generally excellent resource) has a good discussion of restricted constructions and outlines a program for proving the sufficiency of compass-only constructions.
I have started to work through this list. Generally, I won't post solutions since Cut the Knot already has solutions. However, I was concerned about whether the compass they were using was collapsing or not. The solution for 6 (given three points, find a fourth point that makes them vertices of a parallelogram) clearly uses a non-collapsing compass. So, that leaves an open challenge: how to build a non-collapsing compass from just a collapsing compass.
I'm currently stuck on this. There are two ways I could break through:
(a) find the point of intersection of two lines. If I could do this, I would use the ability to find perpendicular lines to move distances around.
(b) Find the point of intersection of a line and a circle with the center on the line.
Note, this second one is very similar to Cut the Knot's third challenge, but that assumes the center of the circle is not on the line segment. Amazingly, this slight change makes the challenge much harder, at least for me.
My running session this morning gave me an idea for a kind of 3-act math discussion with J1 and J2. I will discuss this with them when they come back from camp and see what they think. I expect the last questions will be hard for them and I would like to see how much progress they can make working together.
First Act
Today, I went running and recorded some information on my GPS. For five laps, I ran moderately fast. Here is the data:
Time
Rate
Distance
3:00
12.7 kph
635 m
3:00
12.9 kph
647 m
3:00
12.6 kph
633 m
3:00
12.7 kph
637 m
3:00
12.8 kph
645 m
What do you notice?
What do you wonder?
Second Act
My target was actually to run 12 kph for each of these three minute segments. After the first lap, I knew that I could run more slowly and still hit my target. I wondered, how much less than 635m could I run and still hit my target?
If I compare two laps, both rates and distances, can I figure out the distance I get for each 0.1 kph? Is there another way to calculate the difference in distances for each 0.1 kph?
Third act
For some reason, this made me think about rounding that J1 had recently been studying. He is a bit disturbed about what to do with values that are halfway between the rounded levels, for example whether 15 should round up or down to the nearest ten. Since this investigation of running data involved calculations with measured values and rounding, I though it would be instructive to explore a couple of calculations:
I have two distances, rounded to the nearest 10 cm of 20 cm and 10 cm. What is a reasonable range for the difference of those distances?
My GPS measured a time of 3 minutes (3:00, rounded to the nearest second) and speed of 12 kph (12.0 kph rounded to the nearest tenth of a kilometer per hour). What distance did I run? What is a reasonable range for that distance?
I found several of the puzzles in Gamma pack to be cute, even though they aren't necessarily hard. In especially liked the "reverse" constructions, finding the triangle given the orthocenter (3.2) or given the circumcenter (3.3).
Note: toward the latter half of this pack, I was getting anxious to see what Delta pack had in store, so I bashed through constructions for 3.5-3.8 without always finding the minimal moves solutions. Among those four challenges, I still have 5 missing stars.
Triangle from orthocenter (3.2)
Obtaining the E star gave me trouble on this one. I didn't originally get one, but figured I would try harder for these notes.
The key idea is to use the E-optimal perpendicular construction from 2.6 and the vertex of the angle we're given as one of the center points. That allows us to pick up two perpendiculars for the cost of 5 E moves, leaving one last move to connect the two new vertices.
Triangle from intersection of perpendicular bisectors (3.3)
Well, I actually already gave a spoiler above when I shortened the name of this challenge. If you've got the intersection of the perpendicular bisectors, then you have the center of the orthocircle, the circle that contains all the vertices of the triangle.
Since we already have one vertex and rays where the other edges are....
Three equal distances (3.4)
When going back to write up these notes, I didn't remember how this construction worked and was concerned I'd have trouble working through a tricky challenge. Fortunately, ....
The key insight is the relationship between points B, D, and M. Just think about which of our favorite construction tricks relates them and you are done. Finding E from D and M is straightforward, but keep your eyes opened for a nice surprise!
Those V stars
Three equal distances (3.4), Forty-five degree angle (3.7), and lozenge (aka rhombus 3.8) all have V stars. In fact, 3.8 needs 4 versions to collect the V!
Part of my motivation for writing this series is to make a confession: some of my constructions are just a mess built of geometrically calculating a length that I've determined algebraically. Typically, my approach is to create a coordinate system, then set up a couple of equations that determine a key length for the construction, solve those equations, then use geometric operations to construct that value.
Here is an example. Theoretically, it is a spoiler for a construction in mu pack (circle tangent to two circles 12.5) but I doubt anyone will really be able to see what is going on here:
Another example is construction of the regular pentagon. I know the golden ratio figures prominently, so one approach I often use is just to build that ratio between two lengths and then impose it on the basic construction template.
Other than massively overrunning the move targets for E and L stars, the weakness of this approach is that it doesn't link with any geometric insight about the construction. In a sense, this is the power of analytic geometry: you can get results without having to find a new insight.
Over the course of the series, I'll try to work on constructions that are geometrically insightful, but won't shy away from letting you know where I've had to push through with a brute force attack.
Beta pack is the only one for which we have received all of the stars. Even so, a couple of the puzzles are worth discussing because they illustrate some interesting ideas. In particular, we like Drop a Perpendicular (2.6) and Erect a Perpendicular (2.7).
Perpendicular bisectors
One of the ideas lurking is around the perpendicular bisector, which we saw way back in alpha pack (1.2). One of the things I always found interesting about the perpendicular bisector was that it was easier to construct an object with more conditions than either constructing a perpendicular alone or finding a midpoint alone.
The other great thing about the perpendicular bisector is that it can be defined in an entirely different way: it is the locus of all points that are equidistant from our two starting points. In case that isn't clear, assume we start with two distinct points A and B. The perpendicular bisector of the segment AB is also the locus of all points C such that distance AC is equal to distance BC.
Of course, the fact that this is a line means that we only have to find two such points to construct the perpendicular bisector (which is how you solved 1.2, right?)
Drop a perpendicular
For the 2 move solution using tools, I'll let you find a solution on your own. In case you need a hint: how many combinations of 2 moves are there anyway? You could just try them all and see what you find.
Way back in my HS geometry class, I learned a construction for dropping a perpendicular that uses 4 elementary moves. For the E start, though, that's not good enough. The cool idea that breaks through here is to choose two totally arbitrary points on the line as centers of circles that we draw. For some reason, I get a kick out of the idea that arbitrary points can be helpful ("if the point we choose doesn't matter, how can it help to choose a point anyway?")
In this case, while the points on the line we choose don't matter, the circle we draw with those points as centers need to have the right radius. Click the button below if you want to see how it is done and a bit more explanation.
The key here is that the circle radii are equidistant from our target point and the new intersection point of the two circles. That means they are on the perpendicular bisector of the segment between those points.
Another observation is that the new intersection of the two circles is the reflection of our target point through the line. Kind of cool that, no matter which two center points we choose for the two circles, the new intersection point will always be the same!
Erect a perpendicular
Erecting a perpendicular also uses the idea of choosing an arbitrary point, but goes a step farther. This time, we choose any point we want that is not on the line already! Well, it also fails if we happen to choose a point that is already on the perpendicular, but that should be impossible.... In any case, just choose a point somewhere off to the side.
This construction uses Thales' Theorem. I don't know exactly why I find this result to be so cool, since the proof isn't hard. For me, it transforms a circle into a family of right triangles. Given a length for the hypotenuse, all the right triangles with that hypotenuse are living right there on the arc of the circle with that length as the diameter.
Incidentally, Thales' Theorem and the two defining properties of perpendicular bisectors will come up a lot in other Euclidea Constructions.
V stars
If you want to know where the V stars are, click below:
Angle of 30 degrees (2.3) and Double Angle (2.4) both have two solutions.
Sometime in the past two years, Sue VanHattum, introduced us to Euclid the Game. This is a nice series of classical construction puzzles (compass and straight-edge) built on top of Geogebra. We recently returned to it and saw a link to another version that we've been playing a lot recently: Euclidea.
In EtG, there is a nice discussion in the comments section. Euclidea doesn't have this feature, so I decided to write blog posts chronicling some of our struggles and, hopefully, starting a place for discussion. I'm not planning to write about every challenge or post answers to all of them, but am happy to take requests. Otherwise, I'll write about the puzzles I find challenging and/or interesting for some reason.
Alpha Pack
I think there are two things worth talking about in alpha pack: (1) the puzzle that has stumped us and (2) the location of the V stars.
That darned square (or is it a diamond?)
The last puzzle in the alpha pack is the one that has stumped us. Our target is this inscribed square:
The kicker is to construct it in 7 elementary moves!
Our thought process
We need four line moves to draw the sides of the square. That means we have only 3 moves to find the other three vertices. We can get one by drawing the diameter of the circle, so we have two moves to find the other two vertices.
We can easily find those two side vertices with three elementary moves, but are really stuck on the idea needed to get one less move.
Some spoilers
Six move square
Our approach to get the construction in 8 elementary moves serves easily to get the 6 move construction. Below, I've included the finished picture which should be enough to see the approach (it isn't very involved anyway).
Some quick puzzles created and discussed with J2 while at lunch today. For all, we made the simplifying assumptions that everyone has their birthday on the same day.
Basics/assumptions
Our friends were discussing their current ages: Jin 9, Jate 7, Panelia 6, Sophia 5, and Jane 4. The first two are boys, the latter three girls. All have their birthday on the same day of the year and are having their birthday today!
Add them up
The boys wondered: will it or has it ever happened that the sum of our ages is the same as the sum of the ages of the girls?
Doubles
A little twist on the previous question: when will the sum of the girls ages be twice that of the boys?
For those that know algebra: can you make sense of the solution?
Halves
The previous puzzle was a bit strange. What if we go the other way: when will (or were) the sum of the ages of the boys twice that of the girls?
Murky waters with products
What is the current product of the boys ages? The product of the girls ages?
Have those products ever been equal? We argued this based on the intermediate value theorem.
When were those products equal? We numerically approximated to the closest half year.
Some other families
J2 asked me to include these, which we'd discussed during a dinner last week.
Bill is five years younger than his sister. In seven years, Bill will be 2/3 his sister's age. How old are they now?
John is ten years older than his brother Joe. In six years, John will be twice as old as Joe. How old is Joe now?
What are your favorites?
If you have any of these types of puzzles, please let us know in the comments!
