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Handshakes (Posted on 2004-06-14) Difficulty: 2 of 5
The Smiths, the Andrings and the Cliffords all hold a big party. Everyone shakes hands with every member of the other two families (no one shakes hands with members of their own family), 142 handshakes in all.

Assuming that there at least as many Andrings as Smiths, and at least as many Cliffords as Andrings, how many of each family are present?

  Submitted by Brian Smith    
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Solution: (Hide)
My solution is below. Federico Kereki provides a good solution here.

Let S be the number of Smiths, A be the number of Andrings and C be the number of Cliffords.

From the problem: 0<S<=A<=C and SA + AC + SC = 142.

Combining the two equations gives an upper bound for S: 3S^2 <= 142. S<7.

SA + AC + SC = 142 can be rewritten as (A+S)*(C+S) = 142 + S^2.

Case S=1: (A+1)*(C+1) = 143
A=10 and C=12

Case S=2: (A+2)*(C+2) = 146
No factors give A,C positive

Case S=3: (A+3)*(C+3) = 151
No factors give A,C positive

Case S=4: (A+4)*(C+4) = 158
No factors give A,C positive

Case S=5: (A+5)*(C+5) = 167
No factors give A,C positive

Case S=6: (A+6)*(C+6) = 178
No factors give A,C positive

The only solution in positive integers is S=1, A=10, C=12. There were 12 Cliffords, 10 Andrings and only one Smith.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionPuzzle SolutionK Sengupta2009-02-26 15:28:11
AnswerK Sengupta2009-02-16 16:04:45
Some Thoughtsre: Solution (spoiler)Dej Mar2008-11-03 21:58:15
Hit and trailprashant2004-06-16 08:21:11
re(3): Volume vs surface areaLarry2004-06-15 00:21:09
re(2): 1/2 Surface of a Rectangular Solid...Erik O.2004-06-14 13:26:09
Some Thoughtsre: 1/2 Surface of a Rectangular Solid...Larry2004-06-14 12:29:12
Hints/Tips1/2 Surface of a Rectangular Solid...Erik O.2004-06-14 10:18:20
SolutionNo brute force appliedFederico Kereki2004-06-14 09:58:16
Some ThoughtsONE FOR ALLAdy TZIDON2004-06-14 09:28:28
re(2): Solution - no brute force?fwaff2004-06-14 08:57:15
Some ThoughtsHalf rightLarry2004-06-14 08:24:40
re: SolutionCharlie2004-06-14 08:19:15
SolutionSolutione.g.2004-06-14 07:46:41
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