A new December and a new bunch of puzzles from mscroggs.co.uk.

Let the square be:

abc
def
ghi

Rules:

• (1) def = 2 ghi
• (2) a+d+g = 15
• (3) b+e+h = 19
• (4) cfi = 3 ghi

From these we can deduce these further rules:

• (1a) f is even
• (1b) d >= 2
• (4a) g <= 3
• (4b) c >= 3
• (4c) i is 0 or 5

Assume i = 0. Then by (1) f = 0. Then by (4) h = 0. Then by (1) e = 0. But by (3), then b = 19, not a digit. So, i != 0.

i = 5. Then by (1) f = 0. Then by (4), c05 = 3 gh5, this might be 405 = 3 * 135 or 705 = 3 * 235. In both cases h = 3. g might be 1 or 2. c might be 4 or 7. Then by (1) e = 7. Then by (3) b = 9.

So far we have:

a9c
d70
g35

Assume g = 1. Then by (1) d = 2. Then by (2) a = 12, not a digit. So, g != 1.

g = 2. Then by (1) d = 4. Then by (2) a = 9. Then by (4) c = 7.

The sought after number, abc, is 997.