Answer to May 3, 2004 Problem

The Gambling Problem

 
Pat and Chris are gambling in Atlantic City. When asked to 'put their winnings together,' Pat said that the product was "Nine factorial." Chris added that the sum was "Thirty-eight squared."

How much money did each person win?

 

Solution to the Problem:

This can be done by looking for two numbers m and n ,
whose PRODUCT (as Pat says) is 9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 362880 ,
and whose SUM (Chris adds) is 382 = 1444.
So: m * n = 362880 and m + n = 1444, or n = 1444 - m.

Substituting, we get m(1444 - m) = 362880 ; m2 - 1444 m + 362880 = 0 .
Factoring this is the same as looking for two numbers
whose sum is 1444 and product is 362880.

Factor m2 - 1444 m + 362880 = 0
(m-1120)(m-324) = 0
m = 1120 and m = 324
Then n = 324 and 1120
Pat and Chris had winnings of $1120 and $324.

And if you want to do it without algebra, make a table of pairs of numbers
that add up to 1444, with their products:
Num 1 Num 2 Product
-------- -------- -------
1 . . . . 1443 . . . 1443
2 . . . . 1442 . . . 2884
3 . . . . 1441 . . . 4323
. . . . . . . . . . . . . . . . . . . . . . . . . .
323 . . . 1121 . . 362083
324 . . . 1120 . . 362880 . . YES!



Correctly solved by:

1. Jim Arrison Norristown, Pennsylvania
2. Jeffrey Gaither Winchester, Virginia
3. Richard Johnson La Jolla, California
4. James Alarie University of Michigan -- Flint
Flint, Michigan
5. Saxton Sharad Melrose, Massachussetts
6. David & Judy Dixon Bennettsville, South Carolina
7. Walt Arrison Philadelphia, Pennsylvania
8. Cassie Jordan Harrisonburg, Virginia
9. Mikael Wetterholm Danderyd, Sweden
10. Josh Feingold Winchester, Virginia
11. Celeste Doerwaldt Winchester, Virginia
12. Franklin Harcourt Columbus, Georgia
13. John C. Funk Ventura, California
14. Julie Dallas, Texas
15. Larry Schwartz Trumbull, Connecticut
16. Mark Mendez Harrisonburg, Virginia
17. Chris Maggiolo Harrisonburg, Virginia
18. Monica Quach Harrisonburg, Virginia
19. Kirby Welsko Harrisonburg, Virginia
20. Bella Patel Harrisonburg, Virginia
21. Helna Patel Harrisonburg, Virginia
22. Patrick Mize Columbus, Georgia
23. Kathryn Harris Winchester, Virginia
24. Alex Truong Harrisonburg, Virginia