It stands to reason that all men should finish at the same time. I can't explain why this is true, it just makes common sense.

Let me state up front that I made up this problem myself and I am not 100% positive there is no better answer. If anyone can think of a faster way to cross please e-mail me. That being said I strongly believe that the fastest way to get everyone across is for B and C to start out on foot and A to start out with the bicycle. At a point y A will get off the bicycle and walk the rest of the way. Eventually C will get to the bicycle abondoned by A, then ride backwards to a point x, leaving the bicycle there, then turning around and walk until he reaches the end. Person B will walk until he reaches the bicycle left by C and then ride the rest of the way.

Below are the times that each will take to cross, in terms of x and y:

A: 1*y + 10*(1-y)

B: 5*x + 1*(1-x)

C: 2*y + (y-x) + 2*(1-x)

Next equate these equations: 10 - 9y = -3x + 3y + 2 = 4x + 1.

To solve set up two linear equations:

10 - 9y = -3x + 3y + 2 -> 3x - 12y = -8

10 - 9y = 4x + 1 -> 4x + 9y = 9

Then solve for x and y:

x = 12/25, y=59/75.

Given these points it will take each person 73/25 = 2.92 minutes to cross.

Below are other crossing times, given various crossing times of the two fastest people, assuming the slowest still takes 10 minutes to cross.

Crossing time of fastest -------------------------------------------------------------------- Second Fastest Time 1 2 3 4 5 6 7 8 9 ------- ---- ---- ---- ---- ---- ---- ---- ---- ---- 1 1.00 2 1.62 1.92 3 1.90 2.41 2.74 4 2.06 2.71 3.16 3.48 5 2.16 (2.92) 3.45 3.85 4.16 6 2.23 3.07 3.67 4.13 4.49 4.78 7 2.29 3.18 3.84 4.35 4.75 5.08 5.34 8 2.33 3.27 3.98 4.52 4.96 5.32 5.62 5.87 9 2.36 3.34 4.09 4.67 5.14 5.52 5.85 6.12 6.36

Michael Shackleford, A.S.A., 10/11/1998