Fibonacci Magic For An Old, Wise, Large Man
A man takes a 10-day job. He is told that he'll be paid his boss's age on the first day and his boss's weight on the second day. The third day's wage will be the sum of the first two, the fourth day's wage the sum of the 2nd and 3rd, and each subsequent day's wage will be the sum of the previous two. The man is only told the total payment, for the boss doesn't want to reveal his age or weight.
The man asks his boss if he can see just one of the day's wage (after the 2nd), knowing that he can calculate the age and weight from one day's wage and the total sum.
If the boss wants to conceal his age and weight, which day's wage must he show the man?
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1 solution
Let A = age and W = weight. Then each of the 10 day's wages are as follows: A, W, A + W, A+2W, 2A + 3W, 3A+5W, 5A + 8W, 8A + 13 W, 13A + 21W, 21A + 34W.
The sum of the total = 55A + 88W.
Here we see the total is 11 times the 7th day, thus as a system these two expressions would be linearly dependent and yield infinitely many solutions. Using a system of any other day along with the total would be solvable and meet success for the man.