Question 1206603
<br>
Let the numbers of stamps Alice, Ben, and Carl had originally be A, B, and C, respectively.<br><pre>

    A           B          C     originally

  A-B-C        2B         2C     after Alice gave each of Ben and Carl the number they had

  2A-2B-2C   -A+3B-C      4C     after Ben gave each of Alice and Carl the number they had

  4A-4B-4C  -2A+6B-2C  -A-B+7C   after Carl gave each of Alice and Ben the number they had</pre>
In the end, they each had 80 stamps:<br>
(1) 4A-4B-4C=80
(2) -2A+6B-2C=80
(3) -A-B+7C=80<br>
Simplifying (1) and (2)...
(4) A-B-C=20
(5) -A+3B-C=40<br>
Adding (3) and (4) to eliminate A:
(6) -2B+6C=100<br>
Adding (4) and (5) to eliminate A:
(7) 2B-2C=60<br>
Adding (6) and (7) to eliminate B:
(8) 4C=160; C=40<br>
Plugging C=40 in (7) to solve for B:
2B-80=60; 2B=140; B=70<br>
Plugging C=40 and B=70 in (4) to solve for A:
A-70-40=20; A=130<br>
ANSWER: Ben started with B=70 stamps.<br>
CHECK:
initially: (A,B,C) = (130,70,40)
after first exchange: (20,140,80)
after second exchange: (40,40,160)
after third exchange: (80,80,80)<br>