Problem 25 · 1988 AJHSME
Stretch
Counting & Probability
count-palindromescase-by-digit-count
A palindrome is a whole number that reads the same forwards and backwards. If one neglects the colon, certain times displayed on a digital watch are palindromes. Three examples are: 1:01, 4:44, and 12:21. How many times during a 12-hour period will be palindromes?
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Answer: A — 57.
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Hint 1 of 2
A digital time has a different number of digits at 1:05 (three digits, 105) than at 11:11 (four digits). Reading-the-same forwards and backwards works differently for each, so handle the 1-digit hours and the 2-digit hours separately.
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Hint 2 of 2
For a 3-digit time h:mm, reading backwards forces the last digit of mm to equal h, and the middle digit (the tens of the minutes) is free — but only 0–5, since minutes top out at 59.
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Approach: split by how many digits the hour has
- One-digit hours, h:mm with h = 1–9 (the string is h, then two minute digits). To read the same backwards, the minutes' ones digit must equal h, and the minutes' tens digit is the free middle — it can be 0,1,2,3,4,5 (minutes go up to 59). That's 6 valid minutes for each of the 9 hours: 9 × 6 = 54 times.
- Two-digit hours, hh:mm = 10, 11, 12 (the string is four digits). Reverse must match digit-for-digit, which only works for 10:01, 11:11, and 12:21 — 3 times.
- Total = 54 + 3 = 57.
- Why this transfers: counting under a 'reads the same both ways' rule means free digits in front *force* matching digits in back. Count only the digits you may choose freely (here, the hour and the minutes' tens), and respect each slot's real limits (minutes' tens ≤ 5).
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