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2016/05/31 ...

Some time ago a friend of mine, Adam Atkinson, mentioned to me what he referred to as "Semi-Chestnuts" - puzzles that should be classics, but are for some reason effectively unknown. Recently one of these caught the attention of the Twitter-verse. I've changed it a little here for the purposes of opening the discussion - this is specifically not the version Adam gave me, and is not the version I tweeted - but it is a good place to start the discussion.

So the (modified) puzzle is:

Find three numbers in arithmetic progression whose product is a prime.

When presented with that question, as phrased there, there are two initial reactions. One is to say that it's impossible, because a prime can never be a product. But the other is to start asking for clarification.

  • Do you mean whole numbers?

    • ... and that's a whole 'nother kettle of fish - what exactly is a "whole number"?

  • Is 1 a prime?

  • Can I use negative numbers?

....... and so on.

Until recently when I've been giving puzzles I've always answered those questions, but the resonses from @DrBreitmaul brought me up short. He didn't solve the puzzle as I expected it to be. Well, actually he did, but he went much further.

You see, he asked each of those questions, and then proceeded to solve the puzzle for each of the possible answers. Instead of just taking one branch in the tree of possibilities, he investigated all of them. And the results were fascinating.

So here is a puzzle:

Find three numbers in arithmetic progression whose product is a prime.

Now, ask a clarifying question:

Do you mean whole numbers?

The version Adam originally gave me, and the version I tweeted, explicitly says: Find three integers in arithmetic progression whose product is a prime. Stating it like that side-steps one of the possible explorations, whereas changing to a slightly less well-defined version opens more paths to explore. Including "What is a number?"
Suppose the answer is no, I wasn't intending to restrict you to whole numbers. Can you solve it then? How many different solutions can you find? Is there a structure to all of the solutions? How do you know you've found them all?

Then suppose the answer is yes, you are restricted to whole numbers. Now are there any solutions? How many different solutions can you find? Is there a structure to all of the solutions? How do you know you've found them all?

And so on. So there's a whole new meta-puzzle:

How many different clarifying questions can you find?

How many solutions can you find in each case?

Go ...


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