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This cross-number puzzle has three different solutions that fit the grid. Find them all. In some cases, the same square of the grid can be filled by the same digit in different solutions, but in no case is the complete answer to a clue the same in different solutions. There are no zeros.
1. The sum of the digits is the same as the sum of the last three digits of 6 down.
3. A multiple of 16.
5. The sum of the digits is half the sum of the digits of 7 across.
The first digit is greater than the second digit by the same amount
(which may be 0) as the second digit is greater than the third.
9. Digits are all different, with none greater than 5.
11. A perfect square.
12. Half of 3 across.
2. (Reversed) A multiple of 9.
3. Sum of the digits equals 16.
4. The number formed by the first two digits is twice the number formed by the last two.
Successive digits increase either by the same amount or in the same proportion (i.e., either in an arithmetic or geometric progression).
8. A perfect cube.
10. The magic number.
Find all three solutions and the thus the three different magic numbers (10 down).
The magic numbers are: 12, 24, 31
Derive the three solutions (A, B, and C) as follows:
As a starting point, find some clue or clues for which there are three different answers. You can find this in 8 down and 11 across. Only three, three figure cubes end in a figure in which a square can end (11 across): 125, 216, 729. Place in each of your three solution grids. 11 across of a must be three of 25, B must be 16 or 36 and C must be 49. Solve 4 down and 7 across to determine 11 across for B.
In 4 down, second figure in A must be 4; in B it must be 2 or 6; in C it must be 8. In 7 across, the first figure in A must be 7; B first figure must be 2 and second figure 2 (if second figure were 6, first figure would need to be 10); in C first figure must be 9. First figure of 11 across in B must be 1 and not 3. Fill in the starting diagrams as follows:
3 down A: Second figure of 9 across must be 1, 3, or 4; second figure of 3 across must be 2, 6, or 8. If it was 2, 3 across would be 32 and third figure of 3 down would be 6 (which is not possible). If second figure of 3 across were 6, then 3 across would be 16 or 96. 3 down would then be 178 or 970 and neither are possible. Second figure of 3 across is 8, 4 down is 8442, and 3 across is 48, and 3 down is 475.
3 down B: First figure must be 9 and last figure 5 (first figure of 9 across cannot be greater than 5); 3 across must be 96 and 4 down is 6231.
3 down C: Second figure of 9 across must be 1, 3, or 4; second figure of 3 across must be 2, 6, or 8. If 8 then 3 across would 48, but that is the same as 3 across in A and "in no case is the complete answer to a clue the same in different solutions." If second figure of 3 across were 6, then 3 across would be 16 or 96; 3 down would be 196 or 99_ and it is impossible to make digits add up to 16; second figure of 3 across is 2 and first figure must be 3 and 3 down is 394.
Fill in 12 across in all three solutions. Only one solution in 6 down in all three solutions does not violate 9 across: A=1234; B=1248; C=3456.
From here, the grids can then be completed as shown.