Solution

(related to Problem: Boys And Girls)

There are a good many different solutions to this puzzle. Any contiguous pair, except $7-8,$ may be moved first, and after the first move, there are variations. The following solution shows the position from the start right through each successive move to the end:—

$$\begin{array}{cccccccccc} .&.&1&2&3&4&5&6&7&8\\ 4&3&1&2&.&.&5&6&7&8\\ 4&3&1&2&7&6&5&.&.&8\\ 4&3&1&2&7&.&.&5&6&8\\ 4&.&.&2&7&1&3&5&6&8\\ 4&8&6&2&7&1&3&5&.&. \end{array}$$

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References

Project Gutenberg

1. Dudeney, H. E.: "Amusements in Mathematics", The Authors' Club, 1917

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