AMC 10 · 2006 · #4

Grade 3 arithmetic
Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search
digit-sumplace-valueoptimization greedy-algorithm ↑ Prerequisites: digit-sum
📏 Short solution 💡 2 insights
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Problem
A digital watch shows the time in 12-hour form with AM or PM, for example 9 : 59. The hour is a whole number from 1 to 12, and the minutes are two digits, from 00 to 59. The AM/PM letters and the colon are not digits. What is the largest possible sum of the digits in the display?

Pick an answer.

(A)
17
(B)
19
(C)
21
(D)
22
(E)
23

AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Extreme Principle

The display splits cleanly into an hour part and a minute part, so Tool #7 (Identify Subproblems) lets us maximize each part on its own and then add. Tool #14 (Extreme Principle) drives each part to its biggest digit sum — but the catch is that a bigger clock number is not always a bigger digit sum, so we test the boundary cases instead of just grabbing 12. Tool #3 (Eliminate Possibilities) guards against the trap answer that comes from wrongly picking the 12 o'clock hour.

1STEP 1

Split the display into two parts

The digits sit in two boxes — the hour and the two-digit minutes — and the choices do not affect each other, so maximize each box alone.

total digit sum = (hour digit sum) + (minute digit sum)
2STEP 2

Make the hour digits as big as possible

Hours 10, 11, 12 have digit sums of just 1, 2, 3, so the best hour is a lone 9.

9 → 9, 10 → 1, 11 → 2, 12 → 3
3STEP 3

Make the minute digits as big as possible

The tens digit stops at 5 and the ones digit reaches 9, so 59 is the best minute, summing to 14.

59 → 5 + 9 = 14
4STEP 4

Add the two best parts

The display 9 : 59 adds the best hour to the best minute, 9 + 14 = 23, choice (E); using 12 would trap you at 17.

9 + 14 = 23 → (E)
Answer
23
The display 9 : 59 uses digits 9, 5, 9, and 9 + 5 + 9 = 23, matching (E). No single position can do better: the ones digit is already maxed at 9, the minute tens digit is capped at 5, and no hour beats a lone 9 (since 10,11,12 give digit sums 1,2,3). The biggest possible sum is therefore 9 + 5 + 9 = 23, and it cannot be pushed higher, so choices 17 through 22 are all beaten.
💡Key takeaway

Fatten each spot on its own — the hour 9 (not 12!) plus the minutes 59 give 9 + 5 + 9 = 23.

  • Split the display into two parts
  • Make the hour digits as big as possible
  • Make the minute digits as big as possible
  • Add the two best parts

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