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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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Fit to be tied? Comment 2 of 2 |

I do not believe there is a clear solution to this puzzle. What can be verified is that Lois and Rose were each dealt twelve cards and Dora was dealt eleven. After pairs were removed Lois and Rose still had an even number of cards and Dora had an odd number. As each player still had at least one card, the minimum number of cards in Lois's and Rose's hands is two, giving a maximum number of five cards in Dora's hand.  As Lois and Rose hands are even, they can hold either two or four cards each. As (3) stipulates that the combined hands of Lois and Dora contained more pairs than Rose and Dora, it can be determined that Dora had three cards remaining.

Let the remaining 9 cards be represented as AABBCCDDX.

Situation where Dora was dealt the singleton:
Let Dora hold ABX, Lois hold ABCD, and Rose hold CD.
Satisfying condition (3), the combined hands of Lois and Dora hold the most pairs (the two pairs AA and BB); and the combined hands of Rose and Dora hold the least pairs (zero pairs). [The combined hands of Rose and Lois tie in the number of most pairs, with the two pairs CC and DD].

Situation where Lois was dealt the singleton:
Let Dora hold ABC, Lois hold BCDX, and Rose hold AD.
Satisfying condition (3), the combined hands of Lois and Dora hold the most pairs (the two pairs BB and CC); and the combined hands of Rose and Dora hold the least pairs (the single pair AA). [The combined hands of Rose and Lois tie in the number of least pairs with the single pair DD]. 

Situation where Rose was dealt the singleton:
Let Dora hold ABC, Lois hold ABCD, and Rose hold DX.
Satisfying condition (3), the combined hands of Lois and Dora hold the most pairs (three pairs AA, BB, and CC); and the combined hands of Rose and Dora hold the least pairs (zero pairs). [The combined hands of Rose and Lois hold an intermediate number of pairs, i.e., 1 pair: DD]. 


  Posted by Dej Mar on 2013-04-17 03:48:26
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