Problem 15 · 1994 AJHSME
Hard
Number Theory
patternmod-arithmetic
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Answer: A — Up arrow, then right arrow.
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Hint 1 of 2
The arrows don't go on forever in new directions — watch the picture and you'll see the same 4-arrow shape (right, up, right, down) repeat over and over. A repeating loop means you only care about where 425 falls inside the loop.
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Hint 2 of 2
Sort each starting point by its leftover after dividing by 4: numbers ending the same way in that 4-cycle leave the same way. So point 425 behaves like the 1-leftover points (0, 4, 8 →), and 426 like the 2-leftover points.
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Approach: use the period-4 repetition
- Map the loop using the first numbers: leaving 0 you go RIGHT, leaving 1 UP, leaving 2 RIGHT, leaving 3 DOWN — then it resets (4 acts like 0, going right again). So the arrow out of a point depends only on its leftover when divided by 4.
- 425 = 4×106 + 1, so it has leftover 1 → the UP arrow takes you to 426. 426 has leftover 2 → the RIGHT arrow takes you to 427.
- Up then right is choice A.
- Why this works for any huge number: in a pattern that repeats every k steps, dividing by k and keeping the leftover tells you the position in the cycle. You never simulate all 425 steps — you just locate where 425 lands in one loop.
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