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 Form 21 n Times (Posted on 2016-03-11)
The array below represents a set of 16 playing cards, with the A's representing Aces:

```

A  3  5  8
10  2  4  9
4  2  7  A
2  5  3  8

```

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 March 9.

 Submitted by Charlie Rating: 4.0000 (1 votes) Solution: (Hide) If both Aces were valued at 1, the total of the array would be 74. We can add either 10 (one Ace is 11) or 20 (both Aces are 11). Only 74 + 1*10 is a multiple of 21, that is, 84 = 4*21, and so there are four sets each of which adds to 21. The Ace on the right side of row 3 can't be the 11, as the only way of creating 21 around it would isolate the 8 at the lower right. So the Ace at the upper left must be the 11. Combining that Ace with the 10 immediately below it doesn't allow the rest of the cards to form three more groups of 21. So the only possible division of the array is: ``` A 3 5 | 8 ---+ +---+ 10 | 2 | 4 9 +---+------- 4 | 2 7 A +---+ 2 5 | 3 8 ```

 Subject Author Date Solution spoiler Jer 2016-03-11 11:02:42 No Subject Ady TZIDON 2016-03-11 09:19:53

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