Problem 27 worked answer
The Knight’s Missing Numbers
Designed for Grade 4 · Uses Middle Grade 4 math · About 6–13 minutes
At a glance
The completed outlined quarter is:
| 1 | 2 | 3 | 4 | |
|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 4 |
| 2 | 3 | 4 | 6 | 6 |
| 3 | 4 | 6 | 8 | 8 |
| 4 | 4 | 6 | 8 | 8 |
The knight-move counts that appear are 2, 3, 4, 6, and 8.
The counts that can never occur are 5 and 7.
Key idea or plan
A knight has eight possible L-shaped jump directions when no edge is in the way. For each starting square, check those same eight possibilities and count only the landing squares that stay on the board.
The outlined quarter is enough because reflecting the board across either center line turns every legal knight jump into another legal knight jump. A square and its mirror copy therefore have the same number of legal landing squares.
1. Complete the outlined-quarter map
Start at the outside corner of the outlined quarter and work toward the center. Checking the eight possible L-shaped jumps gives these counts:
- Top row of the quarter: 2, 3, 4, 4.
- Second row: 3, 4, 6, 6.
- Third row: 4, 6, 8, 8.
- Fourth row: 4, 6, 8, 8.
That fills all 16 starting squares in the quarter.
2. Choose every possible count and prove completeness
Looking only at the numbers that occur in the complete quarter, the set is 2, 3, 4, 6, 8. Circle those five numbers. Cross out 5 and 7.
Every square on the full 8 × 8 board can be reflected across the vertical center line, the horizontal center line, or both until it lands in the outlined quarter.
Reflection preserves a knight move: a jump that goes 2 squares in one direction and 1 square sideways still has that same shape after reflection. It also preserves whether the landing square is on the board. Therefore a reflected starting square has exactly the same number of legal moves.
The 16-square quarter map is therefore exhaustive for move counts on the whole board. Since neither 5 nor 7 appears anywhere in that quarter, neither can appear on any of the other 48 squares. The other five stamps really do occur because the quarter map contains at least one example of each.
Check
Mirroring the quarter across both center lines reconstructs the full move-count map:
| 2 | 3 | 4 | 4 | 4 | 4 | 3 | 2 | |
| 3 | 4 | 6 | 6 | 6 | 6 | 4 | 3 | |
| 4 | 6 | 8 | 8 | 8 | 8 | 6 | 4 | |
| 4 | 6 | 8 | 8 | 8 | 8 | 6 | 4 | |
| 4 | 6 | 8 | 8 | 8 | 8 | 6 | 4 | |
| 4 | 6 | 8 | 8 | 8 | 8 | 6 | 4 | |
| 3 | 4 | 6 | 6 | 6 | 6 | 4 | 3 | |
| 2 | 3 | 4 | 4 | 4 | 4 | 3 | 2 |
Only 2, 3, 4, 6, and 8 occur.
Teaching note
A useful extension is to ask how many board squares have each knight-move count. The quarter contains one 2, two 3s, five 4s, four 6s, and four 8s. Each quarter contributes the same counts, so the full board has 4 squares with 2 moves, 8 with 3 moves, 20 with 4 moves, 16 with 6 moves, and 16 with 8 moves.
Technical fit and rating
Discrete mathematics and logic
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