Problem 29 · 2016 Math Kangaroo
Stretch
Number Theory
careful-counting
A date can be written in the form DD.MM.YYYY; e.g. today’s date is 17.03.2016. We call a date “surprising” if all 8 digits used in this notation are different. In which month does the next surprising date occur?
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Answer: B — June
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Hint 1 of 3
All eight digits must differ, so the year YYYY itself must already use four distinct digits.
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Hint 2 of 3
A day's first digit is 0-3 and a month's first digit is 0 or 1, which sharply limits which years can work.
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Hint 3 of 3
Step forward from 2016 to the first year whose digits leave a legal day and month with no repeats.
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Approach: find the first year that leaves room for a valid day and month
- Any year from 2017 onward that starts 20.. reuses the 0 (months and small days also need a 0 or repeat), so no surprising date appears in the 2000s.
- Checking the 2100s, 2200s and early 2300s, the leading digits keep colliding with the only small digits a valid month (01-12) and day (01-31) can use, so none works.
- The first year that frees up enough distinct small digits is 2345, and its earliest surprising date is 17.06.2345 (digits 1,7,0,6,2,3,4,5 all different).
- That date is in June.
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