## Figure Logic (solution)

The clues:

 Across: 1. 11-across squared. 4. 3 more than 14-down. 5. 2 times 15-across. 7. 12-down + 5-across. 9. 2-down - 15-across. 10. 5 more than 9-across. 11. 10 less than 8-down. 13. 1-down + 3-down. 15. 1/2 of 5-across. 16. 4-down - 1-across. Down: 1. 13-across - 3-down. 2. 11-across + 3-down. 3. 30 more than 5-across. 4. 2-down squared. 6. 1-across - 16-across. 7. 10-across rearranged. 8. 3 times 4-across. 12. 2 times 4-across. 14. 20 more than 3-down. 15. 3 more than 15-across.

Here is the solution:

Here is my reasoning, while solving the puzzle (I have a faster method at the bottom):

1. Clue #1A: 1-across is a 5-digit number, so 11-across is between 100 and 316.
2. 12-down cannot begin with a zero, so 11-across is between 110 and 316.
3. 3-down cannot begin with zero, so 11-across cannot end with zero, and 11 across is between 111 and 316.
4. 3-down is 30 more than 5-across, so 5-across must be 11, 22, 33, 44, 55, or 66, and 3-down must be 41, 52, 63, 74, 85, or 96.
5. No square ends in 7 or 8 (or 2 or 3), so 3-down must be 41, 52, 63, or 96, and 5-across must be 11, 22, 33, or 66.
6. 13-across (a two digit number) is 1-down + 3-down, so 3-down cannot be 96, and must be 41, 52, or 63, and 5-across must be 11, 22, or 33.
7. 5-across is twice 15-across, so 5-across is 22, 15-across is 11, 3-down is 52, and 15-down is 14, and 14-down is 72, and 4-across is 75, and 12-down is 150, and 7-across is 172, and 9-across is 256, and 10 across is 261.
8. 1-across ends in 25, so 11-across ends in 5.
9. 11-across is now between 115 and 315.
10. Clue #1D: 13-across is at least ten greater than 3-down, so 13-across is between 62 and 99, and 1-down is between 10 and 47.
11. Clue #12D: 12-down (a three digit number) is twice 4-across (a two digit number), so the first digit of 12-down is 1, and 4-across is between 50 and 99, and 11-across is either 115, 215, or 315, and 1-across is either 13225, 46225, or 99225.
12. In step 10 above, we saw that 1-down is between 10 and 47, so 1-across cannot be 99225, and must be 13225 or 46225, and 11-across is 115 or 215.
13. Clue #16A: The last digit of 1-across is 5, and the last digit of 16-across is 4, so the last digit of 4-down is 9, so 13-across is 97, and 1-down is 45, and 1-across is 46225, and 11-across is 215, and 8-down is 225, and 2-down is 267, and 4-down is 71289, and 16-across is 25064, and 6-down is 21161.

And I think that fills in the puzzle.

The above shows a general method, slowly chipping away at the puzzle. Here is a faster method of solution (selecting our clues more carefully):

1. Clues 3D and 5A show that 5A is 22, 44, or 66, while 3D is 52, 74, or 96.
2. Clue 14D shows that 3D cannot be 96. And clue 1A shows that 3D cannot be 74 (squares don't end in 7).
3. So, 4A=75, 5A=22, 15A=11, 3D=52, 14D=72, and 15D=14.
4. Then, 12D=150, 8D=225, and 11A=215.
5. And, 1A=46225, 2D=267, and 4D=71289.

And the rest is easy.