Henry Ernest Dudeney/Puzzles and Curious Problems/26 - Cross and Coins/Solution

by : $26$

 * Cross and Coins

Solution
The solution given by is:


 * Dudeney-Puzzles-and-Curious-Problems-26-Solution.png

who then goes on to explain:


 * The illustration shows a solution.
 * Only one coin -- the farthing -- is repeated, and in both directions the sum is $\pounds 1, 2 \shillings 7 \tfrac 1 2 \oldpence$


 * Readers may possibly ask how many different arrangements are possible, all correctly adding up to this amount.


 * Well, the penny must, in every solution, remain in its present position, but the seven coins in the upright may be permuted in $2520$ ways.
 * (the two farthings being regarded as indistinguishable),
 * and the four coins in the horizontal may be permuted in $24$ ways.


 * Therefore, $2520 \times 24 = 60 \, 480$ -- the number of different ways in which the coins may be arranged.


 * But we can exchange the three coins $2 \shillings$, $6 \oldpence$ and $\tfrac 1 2 \oldpence$ in the horizontal with the $2 \shillings 6 \oldpence$, $\tfrac 1 4 \oldpence$ and $\tfrac 1 4 \oldpence$ in the upright.


 * This new arrangement will give $5040 \times 12 = 60 \, 480$ solutions also.
 * so the total number of ways is twice $60 \, 480$ or $120 \, 960$.