Enigma 1306: Three all
27 July 2014
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From New Scientist #2464, 11th September 2004
I have in mind three numbers each of three digits (no leading zero) in each of which one digit is 3. Of the following statements about them, three are true and three are false.
(a) The number is a prime.
(b) The number is (appropriately) a cube.
(c) The middle digit is the average of the other two digits.
(d) The third digit differs from the second by 3.
(e) The number has as a factor a two-digit prime the difference of whose digits is 3, or whose sum is a cube.
(f) The number belongs (appropriately) to the set of triangular numbers 1, 3, 6, 10, 15, 21…
What is the sum of the three numbers?