## Smaller Figure Logic (solution)

Here is the solution (The empty puzzle is shown below).

My reasoning:

1. Although it is not yet necessary, we can limit several of the numbers: 1D is between 11 and 33 (clue 6A and 4A doesn't start with a zero). 8D is between 12 and 34 (clue 8D). 8A is between 10 and 39 (8A and 8D start with the same digit. 2D is between 11 and 33 (clue 2D). In fact, 1D and 2D are the same number. 8A is between 13 and 35 (clue 8A). 3A is between 21 and 43 (clue 3A). 4A is between 13 and 35 (clue 4A). Etc.
2. You may notice from the previous steps that many (all, actually) of the numbers are related to each other. Writing every number in terms of 8D, I get:
• 1A=8D+9x(8D-1)=10x8D-9
• 3A=8D+9
• 4A=8D+1
• 6A=3x(8D-1)
• 8A=8D+1
• 9A=(8D+9)x(8D-1)+1
• 1D=8D-1
• 2D=8D-1
• 3D=9A-4A-6 (not simplifying)
• 5D=(8D+5)x(8D+7)
• 7D=8D-1+8D+9=2x8D+8
• 8D=8D
3. From 1A=10x8D-9, we see that the last digit of 1A is always 1. So 2D is in the range 11-19. That makes 8D be in the range 12-20.
4. Knowing that 8D is between 12-20 gives the following ranges:
• 1A: 111-191
• 3A: 21-29
• 4A: 13-21
• 6A: 33-57
• 8A: 13-21
• 9A: 232-551
• 1D: 11-20
• 2D: 11-20
• 3D=9A-4A-6: 213-524
• 5D: 323-675
• 7D: 32-48
• 8D: 12-20
5. 6A ends in 3 or 4 (7D begins with 3 or 4). 6A is a multiple of 3. So 6A=33 or 54, and 8D=12 or 19. That gives us the following table:
• 1A: 111 or 181
• 3A: 21 or 28
• 4A: 13 or 20
• 6A: 33 or 54
• 8A: 13 or 20
• 9A: 232 or 505
• 1D: 11 or 18
• 2D: 11 or 18
• 3D: 213 or 479
• 5D: 323 or 624
• 7D: 32 or 46
• 8D: 12 or 19
6. The left values work, in all cases. For quite a few reasons, the right values do not work. So we can write the left values into the diagram.
7. We could have done the puzzle in fewer steps. With hindsight, we can perhaps see how. For example, we could probably have limited 6A to just a few values without building the table in step 4.

 Across: 1. 8D + 3x6A. 3. 2D + 10. 4. 8D + 1. 6. 3 x 1D. 8. 2D + 2. 9. 3A x 1D + 1. Down: 1. 4A - 2. 2. 8D - 1. 3. 9A - 4A - 6. 5. (4A+4) x (3A-2). 7. 2D + 3A. 8. 1D + 1.