10 March
2019
An interesting puzzle
Read this up in the book “Mathematical Circus”.
You have 2 green balls, 2 yellow balls and 2 red balls. One ball of each color is 11 pounds each. The other ball of each color is 9 pounds each. You have a scale and pan balance. (meaning you can compare the weights of two sides – which is heavier and which is lighter but you never know the exact weight).
What is the least number of weighings required to find out which are the three heavier balls and which are the lighter balls?
Send me personal message with the reasoning.
Answer to the puzzle
Two weighings is all that is required.
Lets call the balls R1,R2,Y1,Y2,G1,G2
Let’s take a pair – say G1 and G2 and split them. On either side add a different color. Say we have R1, G1 one one side and Y1, G2 on another side.
IF one side becomes heavier (say R1, G1), then G1 has to be heavier than G2. Because if G2 was heavier than G1, under no circumstances can R1,G1 be heavier than R2,G2. (At best both sides can be equal in that case).
If you think about Y1 and R1, they can be 9,11 or 9,9 or 11,11. (If they were 11,9 both sides would be equal in above case).
In a second weighing, weigh Y1 against R2.
The possibilities are that Y1 was heavier – which would imply Y1 is 11 and R2 is 9 and therefore we know Y2 and R1.
Or it could be that R2 was heavier – which would imply that Y1 is 9 and R2 is 11 and therefore we know Y2 and R1 again.
Final possibility is Y1 and R2 are same. In which case they have to be 9 and 9. Remember they cannot be 11 and 11 because then Y1 = 11 and R1 = 9 and then since G1 is heavier than G2, in our original weighing R1,G1 could not have been heavier than Y1,G2. So we know Y2 and R1 too.
What if in the first weighing both sides were equal?
This is the easy case. Simply weigh the two balls not used and the whole thing falls in line in sequential logic. So, we weigh R2 and Y2. Say R2 is heavier. That means R1 is lighter. And Y2 is lighter. Which means Y1 is heavier. From the first weighing, since R1,G1 was same as Y1,G2 and we know R1 is lighter, then G1 has to be heavier. That matches with the fact that Y1 is heavier and G2 is lighter (G1 is heavier we found out) and therefore they balance out.