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Old 11-17-2019, 01:05 PM
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Knowed Out is offline
Join Date: Nov 2001
Location: North Kakkalakee
Posts: 15,491

Games Magazine Wild Cards numbers game

The January 2020 edition of Games magazine had a Wild Cards puzzle that I couldn't solve, and the solution they gave doesn't provide the steps taken to arrive at the answer. There has to be a better way than trial and error.

Can you find two three-digit whole numbers a and b such that both the following statements are true?

The difference between a and b is a three-digit perfect square.

The difference between the squares of a and b is either 1 less or 1 more than the product of a and b.

Hint: One of the numbers is prime.
The second statement can be expressed as:

a2 - b2 = ab 1
(a + b)(a - b) = ab 1
(a - b) = (ab 1) / (a + b)

a - b is a three-digit perfect square, listed here: 102 to 312

I ruled out the top three, since the difference between two three digit numbers can't be more than 800. So, we're left with 102 to 282, 19 possibilities for a - b.

If a or b is prime, that leaves 145 possibilities for either, listed here.

At this point, I couldn't figure out any way to simplify or narrow down the possibilities. Going forward, I could only see using trial and error, and there's way too many permutations to slog through.

And so Doper geniuses, I turn to you. What's the proper way to determine the solution? Here it is in spoiler tags, in case this stumps you too.

236 and 377
Difference: 377 - 233 = 144 = 12 x 12
Difference of squares: 377 x 377 - 233 x 233 = 87840
Product: 377 x 233 = 87841
233 is prime
377 = 13 x 29

Note: The number pairs for which the second statement in the puzzle is true are successive numbers in the Fibonacci series.