# Logic Puzzles, Part II

This post is a sequel to Logic Puzzles, Part I. This time I will focus on the ‘Sum Code’ puzzle and consider how to scaffold it for use with pupils. Note: this post will contain some spoilers about solving the Sum Code puzzle so you may want to try solving it yourself before going much further!

## Sum Code – The Puzzle

To give credit where it’s due, this puzzle is from a book called Jumbo Book of Number Puzzles published by Igloo, 2007. I will try and share a better quality image when I can get to a scanner.

## Cracking The Nut

The nice thing about this problem are the constraints within which it works: each letter A to Z represents a unique integer from 1 to 26. The first three clues are:

• A × B = C
• D × E = C
• F × G = C

and these give us quite a lot of information. There is only one number between 1 and 26 that can be expressed as a product in three such different ways. Therefore we know the value of C. Moreover, we know a set of values that must be assigned in some order to A, B, D, E, F and G. (I marked these numbers with a dot underneath them in the lower table to remind myself they were effectively allocated.) The fourth clue is:

• H × H = I

and, with the remaining available numbers, there is only one possible assignment for H and I. And so the game proceeds – indeed, the clues are in quite a neat order to be tackled (approximately) sequentially.

## Scaffolding for Pupils

Chatting with @MissWillisMaths yesterday evening, we debated how best to prepare students for a puzzle like this. It would seem a shame to do the first few steps for them and so we thought about creating a simpler ‘starter’ puzzle to introduce the key logical steps involved.

Certainly it is worth simplifying the puzzle in terms of the number of letters used and, thus, the number of clues. However, we want to retain the principles of different factorisations of a number and perhaps, say, the use of square numbers.

## Creating a Simpler Puzzle

Here is the thought-process I followed to create a 1-10 puzzle:

• Think about the numbers 1 to 10 and their properties.
• None of them can be created from two different products (using 1 in a product may or may not be interesting.. The clue A × B = B tells us something interesting about A but nothing about B).
• There are 3 square numbers.
• There is a cube number.
• Sums to 6, 7, 8, 9, 10 can be written in multiple ways. (Notice in the original puzzle that addition didn’t appear until about a third of the way into the clues.)

Some initial clue ideas:

1. A × A = B
2. C × C = C
3. D × D = E

This gives us that C = 1, A and D are 2 and 3 (in some order) and B and E are 4 and 9 (in the same respective order).

1. E + J = B

This clue now tells us implicitly that E is smaller than B. (Therefore E=4, B=9, A=3, D=2.) And thus we now also deduce that J = 5.

1. G + I = F + J

This narrows down some options but doesn’t fix any further numbers yet. G, I are either 6,7 or 7,8 and F is either 8 or 10, respectively.

1. A × F = E × G

If I’ve done my calculations correctly, that pins down all the remaining numbers to give F = 8, G = 6, I = 7, J = 5.

It’s a little unsatisfying that H has not been clued, but at least we know its value must be 10.

## Starter Puzzle

I’ve summarised the above steps into a single-sheet activity that you could use with a class: SumCode-10. It concludes with a sentence challenging pupils to create their own puzzles which is another great way to get them exploring the deductions involved.

Here are some other sets of clues that might be useful for discussion. This one has a unique solution for puzzles up to 10 (or indeed up to 26):

1. A × A × A = B

This collection of 3 clues would determine B uniquely in a 1 to 10 puzzle:

1. A ÷ B = C
2. D ÷ B = E
3. F ÷ B = G