Henry Ernest Dudeney/Puzzles and Curious Problems/248 - Four in Line/Solution
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Puzzles and Curious Problems by Henry Ernest Dudeney: $248$
- Four in Line
- Here we have a board of $36$ squares, and four counters are so placed in a straight line
- that every square of the board is in line horizontally, vertically, or diagonally with at least one counter.
- In other words, if you regard them as chess queens, every square on the board is attacked by at least one queen.
- The puzzle is to find in how many different ways the four counters may be placed in a straight line so that every square shall thus be in line with a counter.
- Every arrangement in which the counters occupy a different set of four squares is a different arrangement.
- Thus, in the case of the example given, they can be moved to the next column to the right with equal effect,
- or they may be transferred to either of the two central rows of the board.
- This arrangement, therefore, produces $4$ solutions by what we call reversals or reflections of the board.
- Remember that the counters must always be disposed in a straight line.
Solution
There are $9$ fundamentally different solutions:
Of the above, the $4$th, $5$th and $9$th give $8$ solutions when rotations and reflections are considered.
The remaining ones give rise to $4$ each.
Hence there are $48$ different solutions in total.
Also see
Dudeney's Five Dogs Puzzle, no. $311$ in his Amusements in Mathematics.
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $248$. -- Four in Line
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $455$. Four in Line