The array below represents a set of 16 playing cards, with the A's representing Aces:
A 5 5 A
5 2 4 10
3 8 7 4
A A 2 6
Divide the array into sections of adjacent cards so that the sum of the cards' values in each section will be 21. Each Ace can represent either 1 or 11 and you must determine how many sections are needed.
From Page-a-Day Calendar 2016: Amazing Mind Benders, by Puzzability (Mike Shenk, Amy Goldstein and Robert Leighton), Workman Publishing, NY; puzzle for August 6,7.
counting aces as 1, we get a total of 65. We can have up to 4 aces converted to 11 which each add 10 to the total. Of the possible totals 65,75,85,95,105 only 105 is a multiple of 21. Thus we need all aces to be worth 11 and will have to split the grid into 105/21=5 groupings.
So, starting with the Ace in the bottom left corner, it can't join with adjacent ace as that would make a total of 22. Thus it must group with the 3 above it, giving a total of 14 (7 remaining to make 21). Now adding the 8 would again make the total too high, so that leaves the 5 above the 3 and finally we add the 2 to finish the group total at 21.
Now the upper left Ace only has the two 5's on top to work with so that completes a second group.
Moving to the Ace in the upper right, it has to join with the 10 below it to complete a 3rd group.
Now the Ace in the bottom row, second from left, has to join with the adjoining 8 and 2 to finish a fourth group.
This leaves the 4-7-4-6 s-shaped path ending in the bottom right corner for the final grouping.
To summarize, here is a mapping of the groups
2 2 2 3
1 1 5 3
1 4 5 5
1 4 4 5
and based on the deductions above, this is the only possible solution.
Posted by Daniel
on 2016-08-11 09:47:46