Solution to the Problem:
Michael Reilly sent in an excellent analysis of his solution:
Methodology
The $2.05 total tells us that there must be 4 half dollars in the middle row because if we only used (at most) three of them, we would need two other coins to total (at least) $0.55 without a half-dollar, which is impossible. The last coin in that row must therefore be a nickel.
At this point we have five options for that first nickel: any one of the cells in the middle row. So we start making assumptions (I started with the last column and moved right to left). The fifth and fourth columns both led to contradictions, so I'll only follow the middle column assumption branch in this explanation.
The pennies are a great starting point because they are the only coin that can enable a sum to end in any digit other than 0 or 5, so we place the pennies at intersections of column and row totals that don't end in 0 or 5. Since the $0.72 and the $0.67 column totals both need two pennies, we place those first, leaving the last penny in the only remaining such cell.
The second row needs two more coins to reach $0.50, so they must both be quarters.
The first column needs two more coins to reach $0.20, so they must both be dimes.
The bottom row needs 4 coins to reach $0.35, so none of them can be quarters and the remaining 3 quarters must be split between the first and fourth rows. We also have 1 half-dollar remaining that cannot go in the bottom row, so it too must go in the first or fourth row.
-> If we put the half-dollar in the first row, we'll have three cells left that need to reach $0.30. The only way to do that is to use up the 3 remaining dimes. That means we will need to put the remaining quarters in the fourth row, which would make the total $0.77 instead of $0.60. Therefore, the half-dollar does not go in the first row.
We now know the last half-dollar must go in the fourth row, so the two remaining cells in the fourth row must both be nickels. Only the second column can hold the last half-dollar, so the third and fourth columns must hold the two nickels.
This also means that all three remaining quarters go in the first row, and we know they go in the second, third, and fourth columns because a quarter is too large for the fifth column. The last cell in the first row must be a nickel to get the correct total of $0.90.
We know have only to complete the fifth row, which is trivial because each coin is the last in its respective column. To obtain the correct totals, we must place a dime, dime, nickel, and dime in the second, third, fourth, and fifth columns, respectively.
Correctly solved by:
| 1. Cooper Martin |
Mountain View High School, Mountain View, Wyoming |
| 2. James Alarie | Flint, Michigan |
| 3. Skye Grim |
The Field School, Washington, DC |
| 4. Jacob Blumberg |
The Field School, Washington, DC |
| 5. Kelly MacGarrigle | sent from the Tardis |
| 6. Madison Vitt |
Mountain View High School, Mountain View, Wyoming |
| 7. Hannah Bugas |
Mountain View High School, Mountain View, Wyoming |
| 8. Kayle Rippetoe |
Mountain View High School, Mountain View, Wyoming |
| 9. Samantha Brailsford | ---------- |
| 10. Dusty Iorg |
Mountain View High School, Mountain View, Wyoming |
| 11. Chris Stringer |
Mountain View High School, Mountain View, Wyoming |
| 12. Nathon Taylor |
Mountain View High School, Mountain View, Wyoming |
| 13. Marley Newton |
Mountain View High School, Mountain View, Wyoming |
| 14. Hannah Behunin |
Mountain View High School, Mountain View, Wyoming |
| 15. Taylor Meeks |
Mountain View High School, Mountain View, Wyoming |
| 16. Landie Bird |
Mountain View High School, Mountain View, Wyoming |
| 17. Linzy Carpenter |
Mountain View High School, Mountain View, Wyoming |
| 18. Shay Martin |
Mountain View High School, Mountain View, Wyoming |
| 19. Kyla Hoopes |
Mountain View High School, Mountain View, Wyoming |
| 20. Skyler Phillips |
Mountain View High School, Mountain View, Wyoming |
| 21. Blake Murray |
Mountain View High School, Mountain View, Wyoming |
| 22. J. Covington |
Mountain View High School, Mountain View, Wyoming |
| 22. Wyatt Gross |
Mountain View High School, Mountain View, Wyoming |
| 23. Colton Roach |
Mountain View High School, Mountain View, Wyoming |
| 24. Dalton Stoddard |
Mountain View High School, Mountain View, Wyoming |
| 25. hovdej@kpmath.com |
Mountain View High School, Mountain View, Wyoming |
| 26. Michael Reilly | Silver Spring, Maryland |