This blog post is about a game I invented this week, and the game is AWESOME, if I do say myself.

**DIGIT DISGUISES: A game of algebraic deduction**

Players:

- This game is designed for two players, or two teams.

Setting up:

- Each player/team has two grids with the letters A to J, one labelled MINE and one labelled THEIRS, like the picture below.
- Each player/team writes the ten digits 0 to 9, each in a separate box in the MINE grid, keeping the grid where the other player/team can’t see it. (I actually have a printable version with the rules on them that can be turned into battleships-style game stands here.)

The goal:

- Each number has been disguised as a letter. You need to find out which number each of the other player/team’s letters is, by finding out what calculations with the letters produce.

Your turn:

- On your turn, ask for the result of a calculation involving exactly two
*different*letters and one of the operations +, -, ×, ÷. Some examples are A – B or C÷J or H+D or E×G. - The other player/team answers your question truthfully, either telling you which of their letters is the result, or telling you the result is not a letter. To be clear: players
*do not ever say a number*in response to a question. They only ever say a letter or “not a letter”. - You can write notes to help you figure out what you know from the information you have so far.
- Now it is the other player/team’s turn.

Ending the game:

- Once during the game, instead of asking a calculation, you can say you are ready to guess. Then you say what number you think goes with every letter. The other player/team tells you if you are right or wrong.
- If you are right for all letters, you win! If you are wrong for any letters, you lose and the other player/team wins! Either way the game is over.

## Thoughts from Digit Disguises games I have played

This game was inspired by a puzzle I wrote called “The Number Dress-Up Party“. In that puzzle, all of the numbers are at a dress-up party and you have to find the identity of just a few of them by asking them to perform operations. I was reminded somehow of the puzzle and I was wondering about modifications of it. One thing I was wondering was if I had only a few numbers at the party, how long it would take to find out what all of them were, and so this game was born. Later that same day I played it across Twitter with a friend, on either side of a whiteboard at One Hundred Factorial, and with my daughter at home. I was then *completely obsessed. *

I learned a lot from these three early games.

In the first game with Benjamin, I was struck by how quickly the logic got complicated, and then how quickly it all cascaded into finding everything when I finally got a few different numbers. It was interesting thinking about what it meant to get the response “not a letter”. I loved finding ways to keep track of the information I knew so far, and making sure I was using all the information I had so far. The presence of the 0 really made for an interesting ride on Benjamin’s side.

I’ll go first!

A-B— David Butler (@DavidKButlerUoA) September 17, 2019

When I played at One Hundred Factorial, we played in teams on either side of a whiteboard and it was way more awesome! Firstly, it was heaps more fun to play in teams — talking through the logic so far and finding ways to represent it so the rest of the team understood was a really pleasurable experience. I loved hearing other people’s thoughts about what we knew so far and what we should do next. Not to mention having people notice when we had made a mistake in our logic. Secondly, having a huge space to write all of our reasoning was really nice. It was fascinating to see the other team’s approach, which was to have a big grid of which letter could be which number and slowly cross off the possibilities. I had never even considered doing something like that!

Digit Disguises was SO MUCH FUN today at #100factorial! Thank you for playing @zithral @JeremyInSTEM @EmilyLHughes pic.twitter.com/ueVuumJSD3

— David Butler (@DavidKButlerUoA) September 18, 2019

When I played with my daughter C (she’s just turned 11 years old), it was a whole different experience. C had no trouble answering my questions using the letter/number correspondence. In fact, she really enjoyed that part. (Indeed, when I offered to play with C the next day, she was happy just to be the keeper of which letter was what number and let me play Mastermind style.)

I found it rather fascinating that even though they haven’t really done algebra at school, it was perfectly natural for her to refer to these numbers by their letter disguises and to write stuff down on the page in terms of the letters. That is, she didn’t need to be taught about using letters for numbers to really get into the idea of the game. I think that is pretty awesome, actually!

What C was having trouble with was making sense of the information she was getting. She immediately realised that getting something like A-B = “not a letter” meant that A was smaller than B, but she didn’t really know what to do with the information that B-A=C. A little discussion helped her realise that this meant B had to be A+C and both A and C had to be smaller than B. But then it seemed like a huge task to find any letters!

It took a few tries to come up with a representation that was helpful, and a bar model really worked to make sense of it, and even allowed us to pull out information such as I=2F. It was still a little difficult for her to understand how knowing I=2F and I/F=C meant she could know that C=2.

I was thinking that maybe all ten digits was too much for someone her age, but then after finding the numbers 2 and 1, the cascade of finding all the other numbers really gave her a feeling of both power over it, so maybe ten digits was fine..

— David Butler (@DavidKButlerUoA) September 18, 2019

So I reckon that for kids her age, definitely playing in teams is a good idea, or having the whole class be a team against a mastermind, or even the very first time with the teacher against the class as the mastermind with the teacher describing their thought process, to get an idea of the strategies involved. I’m also imagining a version much more like the original dress-up party puzzle where a collection of *students* are the numbers and they know which student is which number and the rest of the class have to figure out which student is which number by asking them to combine with operations.

On the topic of “mastermind”, my friend Alex has created a little python script that will allow you to play the game against it, mastermind-style. If you prefer to play in teams but you only have a few players, then you can play with all of you on one team against the computer!

## Mathematical wonderings

The other thing that is so very awesome about Digit Disguises is how many mathematical wonderings flow out of it so very easily. On top of the usual things that come up during the game, such as what getting an answer of “not a letter” tells you about the numbers involved, you can go a long way wondering about the game as a whole and what happens if you change it.

I have wondered all of these things, but only investigated some of them. I won’t ruin the answers for you for the ones I have thought about already.

- How many questions do you need to finish the game? Is there an algorithm that will finish in the smallest number of questions possible?
- What if the goal was to correctly identify just one number? How quickly can you find the first number?
- Which number is the easiest to find? Which number is the hardest?
- Can you identify all the numbers using only one operation, such as only using subtraction? What is the smallest number of questions to finish the game if you’re only using each operation?
- What if the operations were mod 10? You wouldn’t be able to use subtraction to tell you greater or less than in this case, but would it still be possible to find everything?
- What if you had 0 to some other number, like 0 to 25 or 0 to 5. How long would the game be then? Would your strategy be any different?
- What if you didn’t have 0 or 1? What if you had some negative numbers? What if you had some completely other collection of numbers?
- Is it even possible to find all the numbers if you have a different collection? Are there collections of numbers any size you like where you can’t find any of them? Are there collections of numbers where you can find some but not others?
- For the previous question, if there are only a small number of numbers (like two or three), what are
*all*the sets of numbers that can all be identified?

## Have fun!

I hope you like my game of Digit Disguises. I think it’s *AWESOME! *If you do play it yourself or with your students in a classroom, or you have thoughts about the answers to my wondering questions, or have anything you are wondering about yourself, please do let me know.

I am going to try this when we have a few minutes left in class! How long do you think it will take?

How exciting! At One Hundred Factorial it took us about 20 minutes per game. We were playing with teams of two and three.

“this meant B had to be A+B”

I guess you mean A+C?

Yes. That’s what I meant. I’ve fixed it now. Thanks!

I played this today with a student (rising 6th grade), as well as Numbers and Letters. Both went quite well! He was a rockstar at Numbers, and I was surprised at how often we could hit the target exactly, not just within 10.

For this one, I definitely feel like they picked up some things from asking questions (like they figured out where my 0 was), but agree that they didn’t know how to work through the other information given. But they were able to follow my logic as I talked out loud and solved theirs, so it was still a good time overall.

I have an algorithm that will solve a game in a max of 17 questions, but I’m sure there’s a better way – this one is pretty simple and just relying on finding the 1 using multiplication, then using that to determine all the rest using addition.

Thanks so much for sharing! It’s good to know your experience because I can only get so much from playing myself.

I think I remember at One Hundred Factorial we were made an algorithm that was guaranteed to finish in 14, using multiplication to find 1 and 0 first. Most games seem to finish in about 11 turns when I play.