Solution to the Problem:

There were 15,737 turkeys on the 18th.

Try to work with two unknowns, say A = the 5th number and B = the 6th number.
Then A + B = 79.
B + 79 = B + A + B = A + 2B = 8th,
2A +3B = 9th,
3A + 5B = 10th, and
5A + 8B = 11th = 542.

So solving the system {A + B = 79 , 5 A + 8 B = 542}
gives 5 A + 5 B = 395 , 3 B = 147, B = 49 , then A = 30 ;
we get 30, 49, 79, 128, 207, 335, 542, 877, 1419, 2296, 3715, 6011, 9726, 15737.
My son and my granddaughter were both born on April 18th, so I knew the answer had to be at least 2 turkeys... Just kidding, Mike and Carrie.

Larry Schwartz sent in the following: "This problem is simple if you use the Golden Ratio.
542 * phi^7 = 15737 where phi=(1+sqrt(5))/2."
I didn't think of that, and yet, every year, I assign the following program to my C++ Programming students: Find the first 20 Fibonacci numbers and write the ratio of each number to its successor. The answer approaches the number phi, which is approximately 1.618. Good thinking, Larry!