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Form 21 n Times II (Posted on 2016-08-11) Difficulty: 2 of 5
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.

See The Solution Submitted by Charlie    
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solution | Comment 4 of 5 |

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
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