What is the rule for 7 (the "subtract twice the last digit" method)?

Chop off the last digit, double it, subtract it from the rest, and repeat until you can judge the result.

• 1,729 → 172 − 2·9 = 154 → 15 − 2·4 = 7 → divisible by 7. ✓ (1729 = 7 × 247)
• 343 → 34 − 2·3 = 28 = 7 × 4 → divisible. ✓

It works because removing the last digit and subtracting twice it multiplies the remainder by an invertible factor mod 7. It's fiddly for big numbers, which is why the block method below is often preferred.