🇺🇸 AMC 8 ⇄ switch contest
1996 AMC 8 Stretch

Problem 5

Problem 5 · AMC 8 Stretch Stretch
Logic & Word Problems Counting & ProbabilityNumber Theory account-for-all-possibilitiesorganizing-datapattern-recognitionvisual-representation
A knight starts on the bottom-left square of a standard \(8 \times 8\) chessboard. Each knight move always goes upward, climbing either \(1\) row or \(2\) rows (never going down). Call a move that climbs \(2\) rows 'long' and a move that climbs \(1\) row 'short.' (We don't care about left-or-right, only the rows.) (a) How many rows must the knight climb to reach the top row? (b) List every combination of long and short moves that does it. (c) Stretch: for each combination, how many different orders are there to make those moves? Add them up.
Knight on the bottom-left square of a chessboardNtop row (goal)bottom row (start)
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Answer: (a) 7 rows; (b) 4 combinations (a,b)=(3,1),(2,3),(1,5),(0,7); (c) 4+10+6+1 = 21 orderings
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Hint 1 of 4
The board has \(8\) rows. If the knight starts on row \(1\) (the bottom) and must reach row \(8\) (the top), how many rows does it gain in total?
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Hint 2 of 4
Let \(a\) = number of long moves (2 rows each) and \(b\) = number of short moves (1 row each). The climb adds up to \(7\) rows. Write an equation connecting \(a\) and \(b\).
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Hint 3 of 4
From \(2a + b = 7\): since \(2a\) is even and \(7\) is odd, \(b\) must be odd. List the pairs \((a,b)\) where \(a\) and \(b\) are \(0\) or more.
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Approach: Set up 2a+b=7, list whole-number solutions, then count arrangements
  1. (a) Going from row 1 to row 8 is a gain of \(8-1=7\) rows.
  2. (b) Let \(a\) be long moves (2 rows) and \(b\) short moves (1 row), so \(2a+b=7\). Since \(2a\) is even and 7 is odd, \(b\) is odd. The solutions are \((a,b)=(3,1),(2,3),(1,5),(0,7)\) — 4 combinations.
  3. (c) Count arrangements for each: \((3,1)\) has 4 moves, the single short in any of 4 spots → 4; \((2,3)\) has 5 moves, choose 2 long → \(\dfrac{5\times4}{2\times1}=10\); \((1,5)\) has 6 moves, single long in any of 6 spots → 6; \((0,7)\) → 1.
  4. Adding: \(4+10+6+1=21\) different upward move-patterns.
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