Modular Arithmetic in Puzzles

Modular arithmetic asks only about the remainder left after dividing by some fixed number, called the modulus, and interview puzzles frequently collapse into a one-line remainder calculation once you spot the cycle. Writing "$a \equiv b \pmod{n}$" just means $a$ and $b$ leave the same remainder when divided by $n$ — for instance $17 \equiv 2 \pmod{5}$, since both leave remainder 2.

A common puzzle type: "if today is Wednesday, what day of the week is it in 100 days?" Since the week cycles every 7 days, only the remainder of 100 divided by 7 matters: $100 = 14 \times 7 + 2$, so the remainder is 2, meaning 2 days after Wednesday — Friday. The trick generalizes to any cyclic pattern: clock hours (mod 12), a repeating digit pattern in a large power (e.g., the last digit of $7^{100}$ cycles through 7, 9, 3, 1 every 4 powers, so check $100 \bmod 4 = 0$, landing on the 4th element of the cycle, which is 1), or a sequence of coin flips that repeats every $k$ steps.

The general move in any such puzzle is the same three steps: identify the cycle length, reduce the target number modulo that length, and read off the corresponding position in one cycle — turning what looks like a huge computation into simple division with a remainder.