Question 703332
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Let *[tex \LARGE x] be the amount Alice has just before the final swap, and let *[tex \LARGE y] be the amount that Betty has just before the final swap,  Then, since Carol is going to give *[tex \LARGE x] to Alice and *[tex \LARGE y] to Betty, and retain 8 dollars for herself, she must, just before the final swap, have *[tex \LARGE 8\ +\ x\ +\ y].


So after the final swap, when each will have 8 dollars, Alice has *[tex \LARGE 2x] and Betty has *[tex \LARGE 2y].  Therefore *[tex \LARGE x\ =\ 4] and *[tex \LARGE y\ =\ 4]


Now we can make a table of values going backwards:
<pre>

                     Alice        Betty        Carol
After 3rd Swap:        8            8            8

Before 3rd Swap:       4            4           16
(Because Carol must have 8 left after she gives 4 to each of A and B)

Before 2nd Swap        2           14            8
(Because Betty must have 2 to give to A and 8 to give to C)

Before 1st Swap       13            7            4
(Because Alice must have 7 to give to B and 4 to give to C)

</pre>
John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
<font face="Math1" size="+2">Egw to Beta kai to Sigma</font>
My calculator said it, I believe it, that settles it
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