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.
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).
- 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