The Miller's Puzzle (a multiplication investigation)

who: J1 and J2
what did we use: good old pencil and paper
when: intermittently through the weekend as they recovered from being ill

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Denise Gaskins (of Let's Play Math, you know, one of my top recommendations for parents) included a game from Dudeney's puzzle book: The Canterbury Puzzles. The game is a good one which we will use in an upcoming class, but I won't steal her thunder on that one, so go subscribe to her newsletter and find out for yourself.

I'm always excited to see a new source and had never heard of Dudeney or the book, so I went to look. My suggestion is to skip the introduction and go right to the puzzles. One of the first for our play is the Miller's puzzle. Nine numbered sacks are arranged as below:

You notice that 7x28 = 196, but 34 x 5 is not 196. Can you swap sacks, keeping the 1-2-3-2-1 arrangement, so that the products on both sides are the same number in the middle?

Bonus: those sacks are heavy, so how can you get a solution swapping as few sacks as possible?

Our initial attempts

First, the kids were intrigued and suspicious: is 7x28 really 196 and is 34x5 really not 196?

196 is, of course, an old friend: the square of 14. J2 was really excited to see it appear in this puzzle and the factorization 7 x 28 was a nice complement to his identification of 14 x 14.

Second, their inherent sense of efficiency kicked in and they wanted to see if they could fix the problem by simply changing the three bags on the right side. In itself, that was a good discussion about:

• What are all of the possibilities?
• How do we know, without multiplying, that 43x5 and 45x3 won't be 196?
• What are 43x5, 35 x 4, 53x4, 45x3, 54x3?
• What parity do we get when multiplying an even and an odd? When checking our answers, we can at least look to see if we got the right parity.

Their third step was to try a couple of other simple products with the numbers 1 to 9. Overall, this part of the play had some good conversation and a lot of multiplication practice.

A way forward

I didn't give solutions, but I did ask some questions to help guide the investigation further:

1. Where could the 1 go?
2. Where could the 5 go?
3. Where could the odd digits go?

Don't just take those questions in turn, think about them, then see if your answers to one question give new insight into the others.

Another Idea

I like to look for calculation short-cuts and encourage you to do the same. That gives rise to another mini-hint: what are the relationships between the following number pairs and how can you use that to simplify your search for a solution:

1&2, 2&4, 3&6, 4&8
1&3, 2&6, 3&9

Also, isn't it sad that this cute sequence isn't a solution: 4 x 54 = 216 = 8 x 27? So close . . .

Finally, once you have found solutions, you have opened a new can of worms: how many sack moves does it take to go from the Miller's original arrangement to your solution? How can you tell that you've found the solution with the least moves?

An unrelated picture

A slightly more related picture