Enigma 1162: Triangular or square
20 June 2016
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From New Scientist #2318, 24th November 2001 [link]
Harry and Tom each chose a four-digit number that was either a perfect square or a triangular number and told me its last two digits. I deduced that Harry’s number was one of exactly two four-digit perfect squares or one of exactly two four-digit triangular numbers, but that Tom’s number was one of exactly three four-digit perfect squares or one of exactly three four-digit triangular numbers.
Then they told me that the sum of the digits of Harry’s number was the same as the sum of the digits of Tom’s number.
What were (a) Harry’s number and (b) Tom’s number?