# Solution

(related to Problem: The Exchange Puzzle)

Make the following exchanges of pairs: $H-K, H-E,$ $H-C, H-A,$ $I-L, I-F,$ $I-D, K-L,$ $G-J, J-A,$ $F-K, L-E,$ $D-K, E-F,$ $E-D, E-B,$ $B-K.$ It will be found that, although the white counters can be moved to their proper places in $11$ moves if we omit all consideration of exchanges, yet the black cannot be so moved in fewer than $17$ moves. So we have to introduce waste moves with the white counters to equal the minimum required by the black. Thus fewer than $17$ moves must be impossible. Some of the moves are, of course, interchangeable.

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### References

#### Project Gutenberg

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

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