Circle seats
10 women are sitting around a table. Each women thinks of a number, and tells it to her neighbors. Then each person calculates the avarage of the two numbers of her neighbors (which she heard) and writes down the obtained number.
For example if B 's neighbors are A and C , and A 's number is 2 and C 's number is 0, then B writes down the number .
The figure below shows the written numbers.
If Sarah wrote down the number 6, try to figure out which was the number Sarah thought of!
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1 solution
By definition the average of two numbers A and B = 2 A + B
Define F in terms of H
The average calculated for the women at G is 7 = 2 F + H . F = 1 4 − H
Define F in terms of J
The average calculated for the women at I is 9 = 2 H + J .solve for H = 1 8 − J substitute for H F = 1 4 − ( 1 8 − J ) or F = − 4 + J
Define F in terms of B
The average calculated for the women at A is 1 = 2 J + B . solve for J = 2 − B substitute for J F = − 4 + ( 2 − B ) or F = − 2 − B
Define F in terms of D
The average calculated for the women at C is 3 = 2 B + D . solve for B = 6 − D substitute for B F = − 2 − ( 6 − D ) or F = − 8 + D
The average calculated for the women at E is 5 = 2 D + F . solve for D = 1 0 − F substitute for D F = − 8 + ( 1 0 − F ) or F = 2 − F
2 F = 2
F=1