Section 31.3

Digit Puzzles

Cyclic numbers, digit products, dates with distinct digits, pairwise digit sums, numbers with prescribed digits.

Toolkit

These puzzles translate digit conditions into algebraic equations by place-value expansion, then exploit divisibility, bounds, or combinatorial extremality.

Place value: a two-digit number is $10a+b$ ; a two-digit date/age is $10x+y$ with $0\le y\le 9$ , $x\ge 1$ (or $0$ ).
Digit-sum/divisibility rules: $n\equiv 3n$ ’s digit sum $\pmod 9$ ; $n$ divisible by $5$ iff last digit $\in\{0,5\}$ ; by $3$ iff digit sum divisible by $3$ .
To minimise a number with prescribed digit product: factor the product into digits $2$ – $9$ using as few factors as possible, then arrange ascending; to maximise, use the most digits and arrange descending.
Repeating decimals: $1/p$ for prime $p$ (with $10$ a primitive root mod $p$ ) has period $p-1$ ; multiplying the repetend by $k=1,\dots,p-1$ cyclically permutes its digits.
Shifting a digit between ends: moving the last digit of $N=10x+d$ to the front multiplies by $10^{n-1}$ ; combine with the target relation and reduce to a congruence $10^n\equiv \text{const}\pmod{m}$ , solved by finding the multiplicative order.
A Sidon-type condition (all pairwise sums distinct) among digits from $\{0,\dots,9\}$ bounds the set size by counting the $\binom{k}{2}$ required distinct sums against the $17$ possible sum values.

Problems