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Five Pairs of Balls (Posted on 2016-07-22) Difficulty: 4 of 5
I have five pairs of balls, with each pair being a different color. If two balls of the same color are compared on a balance scale they will balance. Conversely if two balls of different color are compared on a balance scale they will not balance. I also know that the five weights of the balls are in an arithmetic sequence.

With the given information about the balls, devise a strategy to sort the balls into ascending order by weight using only a balance scale at most six times.

No Solution Yet Submitted by Brian Smith    
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colours and orders Comment 1 of 1
It's not clear to me if the puzzle asks for a device that sort the balls following a particular order or just to sort them in order of weights. 

If the aim of the puzzle is the second a possible way could be this: 

a) 1st to 3td weights: take three balls of different colors and compared them pairwise. This take three uses of the balance and you get three balls in order.

For ex, suppose order is: Y * G * R (B,W unused)  (§ means heavier than; * means lighter than:

This left 20 possibilities of order for the five colours.

b) The 4th weight could be: two yellow balls (the lighter one) versus the green (the second of the three chosen) (YY vs G).

Three possibilities: 
  • YY § G  (this cover 6 possibilities) [_YGR_][_YG_R][__YGR]  (Group I)
  • YY = G (this allows 8 possibilities) [YGR__][YG_R_][YG__R][_Y_GR] (Group II)
  • YY * G (this cover the other 6 possibilities) [Y__GR][Y_GR_][Y_G_R] (Group III)

c) The 5th weight will be different of the each of the three groups
Group I: YY versus R. 
  • YY § R the pattern will be [__YGR]. Then in 6th weight B vs W determine if [BWYGR] or [WBYGR]
  • YY = R the pattern will be [_YGR_]. Then in 6th weight B vs W determine if [BYGRW] or [WYGRB]
  • YY * R the pattern will be [_YG_R]. Then in 6th weight B vs W determine if [BYGWR] or [WYGBR]

Group II: YB versus R (Use here B for the first time)
  • YB § R the patterns of order will be [YG_RB][YGRB_][YGR_B]. Then in 6th weight YYG vs W determine which of this three.
  • YB = R the patterns will be [YGBR_][YG_BR][_YBGR]. Then in 6th weight YG vs W determine which of this three.
  • YB * R the patterns will be [YGB_R][BY_GR]. Then in 6th weight YG vs W determine which of this two.

Group III: B versus W (Here is better determine which of then is heavier). We will use the heavier (suppose W) to sort this group.

In 6th weight YG versus W
  • YG § W pattern of order will be [YBWGR];
  • YG = W pattern will be [YBGWR]; 
  • YG * W pattern will be [YBGRW]
If the heavier ball in weight 5th was B, B and W permute.

Edited on July 25, 2016, 2:08 pm
  Posted by armando on 2016-07-25 13:00:59

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