Missing Number

When the rule lives inside the digits

A large family of missing-number puzzles never add the values — they operate on digits: sums of digits, reversed digits, products of digits. Recognising the family early saves minutes.

The digit-sum rule. $S(n)$ = sum of a number's digits: $S(47) = 11$. Typical rules:

• third = $S(a) + S(b)$ — e.g. 47 and 38 give $11 + 11 = 22$;
• third = $S(a + b)$ — add first, then sum digits (68 → 14);
• third = $2 \times S(a)$ — doubling a digit sum. Tell them apart by testing one complete row with each variant; only one will fit both rows.

The reversal family. $R(n)$ = the number with digits reversed: $R(43) = 34$. Common rules:

• third = $R(a) + R(b)$ — the classic 'reversed-sum' grid (34 + 41 = 75);
• third = $|R(a) - b|$;
• third = $R(a + b)$ — add first, then flip (39 → 93). Reversal rules produce third numbers whose digits are the 'mirror image' of the row's story — options like 75 next to 43 and 14 are a waving flag.

The digit-product rule. $P(n)$ = product of digits: $P(47) = 28$. Rules: third = $P(a)$, third = $P(a) + P(b)$, third = $P(a) \times S(b)$. Zero is the instant tell — a row containing 0 or 10 has digit product 0, which usually kills product rules on sight.

Spotting the family. Hints that the rule is digit-based:

• The third numbers are far too small or too large for plain sums/products of the values.
• Options cluster near digit-scale numbers (5-30) while the values are two-digit.
• The same digits recur across rows (43/34, 62/26) — reversal is in play.

Verification is cheap here. Digit rules are one line of arithmetic per row — check all complete rows, not just one. And compute the final answer with the SAME variant that fit, not with the variant you tried first.

The mod-9 shortcut. A number and its digit sum leave the same remainder on division by 9. So if a digit-sum rule is in play, the output and the inputs' digit sums must agree mod 9 — a quick way to kill an option without computing every digit sum. Example: S(47) + S(38) = 22; check 47 + 38 = 85 → 85 and 22 both leave 4 on division by 9 ✓.

Reversal traps. Two traps dominate: reversing the inputs when the rule reverses the result (and vice versa), and sums ending in 0 (R(40) = 04 = 4) — setters avoid them, so a 0-ending sum usually means you picked the wrong variant. Palindromes (77, 66) sitting in the grid are a gift: they are identical under reversal and hint that reversal is part of the rule.

Quick revision

• $S(n)$ = digit sum (47 → 11); $R(n)$ = reversal (43 → 34); $P(n)$ = digit product (47 → 28).
• Standard rules: $S(a)+S(b)$, $S(a+b)$, $2S(a)$, $R(a)+R(b)$, $|R(a)-b|$, $R(a+b)$, $P(a)+P(b)$.
• Test each variant on a complete row; only one fits all rows.
• Options far smaller than the values → digit rule; mirror-looking pairs → reversal.
• Zeros kill digit-product rules instantly.