## The Activities

**Topic: Probability:**Book: A Very Improbable Story by E. Einhorn.**Topics: Probability, Pigeonhole Principle:**First I asked some problems similar to those in “A Very Improbable Story”, such as “What’s the chance that two random socks picked out of a drawer will match match, if the drawer has 40 socks, 10 each in 4 different colors?” Next I asked some simple pigeonhole principle question about the same drawer, such as “If you wanted to get a matching set of horseshoes (socks) for a pony, how many horseshoes would you need to be sure you got one?” The final hardest question was “If you have a drawer with 400 horseshoes, 100 each in 4 different colors, and you had 4 ponies, how many would you need to draw to guarantee a complete set for each pony (not necessarily of different colors)?”**Topic: Codes:**I made a letter substitution code (go here to make your own) with a going-away letter to one of our kids who is moving to India, pictured below. The kids had to solve it without a key. Note that the full letter started with “Dear XXXX”, which is important because their first foothold into the code was guessing that it would mention XXXX.

**Topics: Sudoku, Logic:**We did a Unifix Soduku puzzle as a group.

## How Did It Go?

All six kids attended. This week was the going-away party for one of our math circle kids, who is moving to India. She made us this card:

#### A Very Improbable Story

This was a pretty entertaining book and had several interesting probability computations.

#### Sock Drawer

There were a lot of wrong guesses for all the questions. They needed step-by-step guidance in order to answer the probability questions, but once I led them through it, they were able to answer (although still not right away): “After the first sock, how many matching socks are in the drawer?” “How many socks are in the drawer?” They did understand that this meant that the probability was “X out of Y”, perhaps definitionally (this is a pretty decent of probability though, so that’s fine).

Similarly, they didn’t get the pigeonhole principle questions right away, although we’ve done similar in the past. With some help they solved all the simple ones. The final question was quite a bit harder, but they did get the right idea in the end and answered the question (with me doing the “bookkeeping” on a piece of paper). The colors of the horseshoes were gold, silver, black, and clear. For solving the hard question, the kids all got a small prize at the end of circle.

#### Letter Cipher

It took them quite a while to get started. They noticed apostrophes in various places, and even guessed “don’t” for one of the words, but didn’t actually try out whether it looked promising. Eventually one of the kids suggested it might be about kid XXXX, but even then it took them a while to decide to try finding a word of the right length. Once they had written in XXXX, I encouraged them to put all the matching letters in other places to see if it worked. Only some of the kids were engaged, and there were a number of times when someone wrote in a random guess, which I discouraged (except in the rare case where they guessed correctly, just to keep it moving along). It was also quite loud during this activity. The kids aren’t that great at the “algorithm” of letter ciphers, which is, determine some word you know, find all the instances of the numbers in that word, and then look for another word you know. They were scattershot about which letters they filled in. In the end, they did solve the whole puzzle, which mysteriously mentioned logic puzzles; I then gave kid XXXX a present of Mindware Math Perplexors: Basic Level (a book of logic puzzles).

#### Unifix Sudoku

The kids were definitely able to answer questions like “In this region, where does the red square have to be?” So with me picking the questions, we solved it pretty quickly. Of course, to solve a Sudoku, you also need to have a good search strategy to ask the right questions, which is the next step. Amazingly, we made it through the whole puzzle without anyone destroying board.