Is John a liar? and other good puzzles

who: J1 and J2
what did we use: Smullyan's Alice in Puzzle Land and some online games
when: after dinner

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This post is basically a thank-you to all the people who create wonderful books, puzzles, and games. Also, thanks to the people who spread the message by telling us about these resources. In this case, we're directly indebted to A O Fradkin, Annie at the Math Forum, Mike Lawler for these specific activities.

Alice in Puzzle Land

The title of this post comes from one of the opening puzzles in Smullyan's book of logic puzzles: Alice in Puzzle Land. Here's how it goes:

John, and his brother, have the peculiar characteristic that one always tells the truth and the other always lies. Unfortunately, we don't know which one tells the truth and we don't know which one is John. Gosh, we don't even know John's brother's name! What single yes/no question can you ask one of the brothers to figure out which is John? 

What if we want to figure out whether John always tells the truth or always lies, what single yes/no question can we ask one of the brothers to determine this?

To be honest, I didn't get the answer, but J2 managed to figure it out after they posed a bunch of questions and we worked through the possibilities together, example "what if the one we are asking is John and he tells the truth?"

The Js (mostly J1 and J2) have had a lot of fun discussing this particular challenge and have also really enjoyed the tart-ingredient thieves series. They are currently working on the "very complicated logic puzzle" at the end of the first chapter to determine whether the Gryphon or the Mock Turtle was the tart thief.

SolveMe Mobiles

Since we started teaching the math games class at school last year, I have had a strong bias away from on-line games and puzzles. Part of this is simply that we wouldn't be able to use these puzzles in the class, but also because I wanted to help parents see that they don't need any special tools to help their kids have deep mathematical experiences.

Anyway, the SolveMe Mobiles and Game About Squares (below)  helped me recognize and reject this bias!

The mobile puzzles are from EDC, a group with which I am not affiliated, but which I hold in high regard. Each puzzle gives you a balanced mobile that, secretly, codes equations to solve for the weights of the various shapes. J2 (5yrs old) has especially enjoyed this and has just finished puzzle 51:

Even J3 (3yrs old) has gotten something out of this puzzle. She was looking at the screen and we (J0, J1, J2) asked: what do you see, what do you notice, what do you wonder (not all at once.) She talked about shapes, she talked about how many, she asked questions about what different things meant, she tested putting numbers into the blanks.

Game About Squares

Another cool puzzle: Game About Squares. I don't know who to thank for creating this and making it freely available, but it was part of a really nice Notice and Wonder post by Annie which is, itself, great and you should go read it. In the spirit of the game, I'm not really going to tell you much about it, just go play!

Bedtime math extensions

Who: J1 and J2
Where: all around the house
When: 6 am

It is summer vacation for J2 and J3, so we have been doing a lot of activities. Unfortunately, there is almost no free time to blog about it.  For now, let me share some of the conversations J1 and I have had about recent Bedtime Math posts.

Doubling Volcanoes

Today, we talked about this post: Instant Island. In particular, we focused on the last part, the "big kids bonus" question: if an island is ¹/₂ mile wide and the width doubles every month, how wide will it be in 4 months?

Okay, forget about the answer for a moment, does the question make sense? First, we asked: if this were true, how long would it take the volcano to stretch around the circumference of the Earth? Using some familiar powers of 2, we figured out it would be slightly less than 16 months. It seemed pretty clear that this didn't make physical sense, since we clearly don't see small islands cover the Earth like that.

We talked about this doubling growth process for a while. What things do we know that work like this? J1 listed:

• bacteria
• people
• computer viruses
• plants

We talked about what is happening: basically, the new "material" is able to reproduce itself, so the more you have, the more productive potential you have.

Does the growth keep going forever? No, otherwise everything would be covered, actually become, the thing that is doubling. At some point, these all run into a limiting factor, food, water, space, etc.

Back to the island: J1 realized that the growing island would hit other landmasses before it went around the Earth. If you consider connecting with Asia to count as "growth" for the island, then there could be moments of extremely fast growth. At the same time, we know that the Eurasian landmass isn't growing with an exponential process, so this connection won't contribute to the further growth of the volcanic island.

Stuffed with lead?

We were scratching our heads about the weights in this Stuffed Animal post. The average weight per stuffed animal assumed in this post is 7 pounds. We had two conversations about this: how much is that in grams and how much do our stuffed animals actually weigh?

First attempt, a scant 27 grams

Hefty Panda is only 281 grams

Massive Diplodocus is about 100 grams lighter

One of our heavy-weights: still under 400 grams

Not a stuffed animal. This is the trickier we had to use to break 1 kg

Suffice it to say, we had no stuffed animals that exceeded 1 kg.

Heavy Pencils

After having an experience with unreasonable weight measurements, Pointy Gorilla helped launch a similar conversation. Actually, it came from misreading! We connected the following comments:

1. Gorilla weighs 300 pounds (we read this as the pencil gorilla)
2. The big kids question implies a certain number of pencils used (under 600)

So, how heavy are those pencils? Given what we know of real pencils, how much would that gorilla actually weigh?

Grand Catch-up

Who: everyone in the family
When: over the last week
What material did we use: see below, a lot of different goodies
Where: all over the house

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We haven't posted many activity summaries recently, so this is a review of what we have been doing so that you don't think we've been slacking off.

Calculating and Roman Numerals
Remember my doubts about roman numerals when they were introduced (here)? P has since found that they come up frequently when J1 and J2 are discussing some other calculation (adding 2 digit numbers without paper and pencil was a recent one).  It seems that the kids like the process of conversion or the feeling that they are speaking in a type of code.

Mathsemantically, they appear to appreciate the idea that number names are not the same as number concept. Frankly, I don't know when they developed this appreciation as we've always been a bilingual household (Thai and English) and introduced counting in other languages (Spanish, German, Chinese, Korean, Tagalog, Armenian) when they were very young.

Tip: for kids around 7 years old, try introducing the numbers in a different language and see if they enjoy the disassociation of names and objects as they play with calculations.

Food Math
We've had another round of pizza dough (with J3, our 2 year old) and our first attempt at gougeres (J1 and J2).  The pizza dough with J3 went as usual: a lot of fun measuring, a bit of flour outside the mixing bowl, excessive use of the scale to weigh whatever she saw.

The gougeres were a dairy indulgence, with milk, butter, and cheese as main ingredients. This was my first opportunity to try them since J3, our dairy intolerant one, was out for the day with mommy. We roughly followed this recipe, halved because the remaining munchkins had already decided they wouldn't like them and I couldn't justify eating 20-30 cheese puffs myself (and I'm not generous enough to share them with anyone else). In any case, by the time I thought to snap a picture, we only had 5 left for a nice little pentagonal arrangement.

Mathematically, the activity was interesting because we halved the recipe and we had a debate about how many eggs to use. While the presenter explicitly says 5 eggs, both boys were sure we only saw her use three and we all agreed that she had reserved one for the egg wash. Coincidentally, P brought home some very small eggs later in the day, so we got to have a little (very little) discussion of whether one egg is a proper measure. I'll have to remember to return to that again.

As we've hinted on other occasions, we try to recognize little unplanned opportunities for inserting some numerical discussion into everyday conversation.  Quesadillas at lunch were a chance to talk about fractional parts of a circle based on 1/8.  Below is one serving in the 6/8 (aka 3/4) uneaten state:

Some bites and a cheeky smile later brought us: "daddy, what about this one?"

I offered 1/3rd of an 1/8th for the partially eaten piece and we calculated that this plate still had 7/24th. Maybe I should have rushed to get a clock and show them 3.5 hours?

Catan (Catan, Catan)
Somehow, Settlers of Catan got back on J1's radar and he's been asking to play it everyday for a while. We are still working on a version that fully integrates J2 and it is a bit stale without trading between players.  I tried to introduce a bit of trading today, but J1 was too suspicious of my motives and would rather trade at an unfavorable rate with the bank than trade with me.  For now, we still start him with 2 cities against my 2 settlements, and we also allow all the players to start with the resources from all the hexes associated with both of their developments.  Otherwise, we find the game takes too long to get moving.

You can see J1 (red) in the process of beating me (orange) below:

I can tell these sessions have started to develop his intuition for dice probability.  When allowed to set up a board configuration of his own choosing, he put the 6s, 8s and 9s together on the hexes associated with his developments.

And some geometric designs from J2 while he was watching us play:

Hexaflexagons
I plan to have more discussion of this tomorrow after our math party. For now, suffice it to say that I made some flexagons, left them around the house, and both the boys were really excited to make, decorate, and investigate them.  Almost all of them are trihexaflexagons, so that's what they've come to expect.  I gave J1 a hexahexflexagon with numbered faces and he was delighted to discover the extra faces.  I'm looking forward to seeing how the other first graders respond.

Since it isn't fair to entirely leave you hanging, I'd refer you to this Vi Hart video for a great intro to flexagons: http://www.youtube.com/watch?v=VIVIegSt81k