Henry Ernest Dudeney/Modern Puzzles/138 - The Four Draughtsmen/Solution
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Modern Puzzles by Henry Ernest Dudeney: $138$
- The Four Draughtsmen
- The four draughtsmen are shown exactly as they stood on a square chequered board --
- not necessarily eight squares by eight --
- but the ink with which the board was drawn was evanescent,
- so that all the diagram except the men has disappeared.
- How many squares were there in the board and how am I to reconstruct it?
- I know that each man stood in the middle of a square,
- one on the edge of each side of the board and no man in a corner.
Solution
Construct $AD$.
Construct $CE$ perpendicular to $AD$ and make $CE = AD$.
Because $AD$ stretches from one side of the board to its opposite, joining the centres of two squares, then the same applies to $CE$.
So $E$ is the centre of a square on the same row as $B$.
So join $EB$ and produce it in both directions.
Then:
- construct $FG$ through $C$ parallel to $EB$
- construct $FH$ through $A$ perpendicular to $EB$ and $FG$
- construct $DG$ through $D$ parallel to $FH$.
As $H$ is the centre of a square on a board, we can mark $HE$ all round the square $HFG$ and discover the board is $10 \times 10$.
It remains to construct the actual board itself.
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $138$. -- The Four Draughtsmen
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $265$. The Four Checkers