Enigma 1741: Four squares
20 March 2013
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From New Scientist #2909, 23rd March 2013 [link]
I have before me five two-digit numbers, with no leading zeros. All the digits are different and none of the numbers is prime. The sum of the five numbers is a perfect square. If I remove one of the numbers, the sum of the remaining four is also a perfect square. If I remove another number, the sum of the remaining three is again a perfect square, and if I remove a third number, the sum of the last two is again a perfect square.
What are my five numbers?