Hello Math-fun, We've started a spiral of distinct integers with 0 in the middle of the plane, then 1, then 2. At this point the next integer must be 6 as we want the content of EVERY 2 x 2 SQUARE of the plane to sum up to a square (0+1+2+6=9). When we have a choice, we always extend the spiral with the smallest unused term. Thus here, after 6, the term 3. This 3 forces us to fill the next cell with 7 (indeed 0+6+3+7=16) and not 8 (the sum 17 is not a square) or 40 (though 0+6+3+40=49 is also a square) -- remember, we must always extend the spiral with the smallest unused integer. This means that the only constraint on the next integer (after 7) is to be the smallest unused one, thus 4. Now to complete the next 2 x 2 square we must use 5 (as 0+7+4+5=16), etc. +---+---+---+---+---+---+---+---+---+---+ | | | | | | | | | | | +---+---+---+---+---+---+---+---+---+---+ | | | | | | | | | | | +---+---+---+---+---+---+---+---+---+---+ | | |13 |38 |34 |15 |31 | A | | | +---+---+---+---+---+---+---+---+---+---+ | | |26 | 4 | 5 |10 | 8 | B | | | +---+---+---+---+---+---+---+---+---+---+ | | |12 | 7 | 0 | 1 |17 | C | | | +---+---+---+---+---+---+---+---+---+---+ | | |14 | 3 | 6 | 2 |16 | . | | | +---+---+---+---+---+---+---+---+---+---+ | | |11 |21 |19 |22 | 9 | . | | | +---+---+---+---+---+---+---+---+---+---+ | | | | | | | . | . | | | +---+---+---+---+---+---+---+---+---+---+ | | | | | | | | | | | +---+---+---+---+---+---+---+---+---+---+ This spiral leads to the lexicographically earliest sequence of distinct positive terms 0, 1, 2, 6, 3, 7, 4, 5, 10, 8, 17, 16, 9, 22, 19, 21, 11, 14, 12, 26, 13, 38, 34, 15, 31, A, B, C, ... which is not in the OEIS and perhaps of general interest (caveat: handmade, as usual). (Our next move after dinner, soccer on TV, sleep, breakfast? Each 2 x 2 square sums up to a prime, of course). Best, É.