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Singleton Settlement (Posted on 2013-04-16) Difficulty: 3 of 5
Dora, Lois, and Rose played a card game with 35 cards, consisting of 17 pairs and one singleton.
  1. Dora dealt one card to Lois, then one card to Rose, then one card to herself, and repeated this order until all the cards were dealt.
  2. After the pairs in each hand were removed, at least one card remained in each hand; the number of cards in the three hands totaled 9.
  3. Of the remaining cards, Lois' and Dora's hands together formed the most pairs, and Rose's and Dora's hands together formed the least pairs.
Who was dealt the singleton?

No Solution Yet Submitted by K Sengupta    
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Solution solution | Comment 1 of 2

The 9 remaining cards mentioned in item 2, are the one singleton and four of the original 17 pairs. By the way that the cards were dealt, Lois and Rose started out with 12 cards each while Dora had only 11.

Rose had the least number of cards from the remaining four pairs while Dora had the second least (from item 3).

My attempt at a rectilinear Venn diagram is below. Since there are four pairs remaining (each divided between two ladies), the three intersections are marked with numbers that add to 4 but have a definite order from least to most. That's only possible with 0, 1 and 3 as the numbers, so Lois and Dora share the most: 3 pairs between them, each having three cards as the three half-pairs. Rose and Dora share the least, that is, none; Lois and Rose have the one pair to share between them, so Lois has 4 of these cards, and Dora has 3.

Dora was the only one who had an odd number of cards to begin with, and since cards were discarded in pairs, she's the only one left with an odd number of cards. The cards we mentioned already, the three that have counterparts in Lois's hand, are odd in number, so she can't have the singleton as that would make it even.

Lois and Rose started out with an even number of cards to begin with, and also to end with. Lois's three that are counterparts with Dora's three plus the one she has in a match with Rose make for the even number 4. But Rose has only one card yet accounted for, the one matching one of Lois's cards, so she needs the singleton to make her card count 2, which is even, as she needs it to be.

So Rose has the singleton, as noted by the 1 in the Rose-only area of the Venn diagram.

(There were no 3-of-a-kind sets, so the central part of the Venn diagram shows this as 00.)

               Lois         Rose
             +--------+  +--------+
             |        +--+        |
             |        | 1|   1    |
             |    +---+--+---+    |
             |    |   |00|   |    |
             |    | 3 +--+ 0 |    |
             +----+---+  +---+----+
                  |          |
                  |          |
                  |          |

  Posted by Charlie on 2013-04-16 11:45:47
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