# Weighing Pool Balls Puzzle - Solution

### The Puzzle:

One of twelve pool balls is a bit lighter or heavier (you do not know) than the others.

At least how many times do you have to use an old balance-type pair of scales to identify this ball?

At least how many times do you have to use an old balance-type pair of scales to identify this ball?

### Our Solution:

It is enough to use the pair of scales just 3 times.

We know of two possible solutions:

Let's mark the balls using numbers from 1 to 12 and these special symbols:

x? means I know nothing about ball number x;

xL means that this ball is maybe lighter then the others;

xH means that this ball is maybe heavier then the others;

x. means this ball is "normal".

At first, I lay on the left pan balls 1? 2? 3? 4? and on the right pan balls 5? 6? 7? 8?.

If there is equilibrium, then the wrong ball is among balls 9-12. I put 1. 2. 3. on the left and 9? 10? 11? on the right pan.

If there is equilibrium, then the wrong ball is number 12 and comparing it with another ball I find out if it is heavier or lighter.

If the left pan is heavier, I know that 12 is normal and 9L 10L 11L. I weigh 9L and 10L.

If they are the same weight, then ball 11 is lighter then all other balls.

If they are not the same weight, then the lighter ball is the one up.

If the right pan is heavier, then 9H 10H and 11H and the procedure is similar to the former text.

If the left pan is heavier, then 1H 2H 3H 4H, 5L 6L 7L 8L and 9. 10. 11. 12. Now I lay on the left pan 1H 2H 3H 5L and on the right pan 4H 9. 10. 11.

If there is equilibrium, then the suspicious balls are 6L 7L and 8L. Identifying the wrong one is similar to the former case of 9L 10L 11L

If the left pan is lighter, then the wrong ball can be 5L or 4H. I compare for instance 1. and 4H. If they weigh the same, then ball 5 is lighter the all the others. Otherwise ball 4 is heavier (is down).

If the left pan is heavier, then all balls are normal except for 1H 2H and 3H. Identifying the wrong ball among 3 balls was described earlier.

This solution was provided by Charles Naumann. His method also solves it with just three weighings:

Label the balls 1-12

First Weighing:

Left: 1 2 3 4

Right: 5 6 7 8

Off: 9 10 11 12

Record the heavier side (L, R, or B)

Second Weighing:

Left: 1 2 5 9

Right: 3 4 10 11

Off: 6 7 8 12

Record the heavier side (L, R or B)

Third Weighing:

Left: 3 7 9 10

Right: 1 4 6 12

Off: 2 5 8 11

Record the heavier side (L, R, B)

There are 27 (3^3) possible combination of scale readings. A complete sorted list of the scale reading appears below. Note that only 24 of the 27 readings should be possible given the original problem statement. The algorithm was designed so that if all three scale readings are the same, an error is flagged indicating that the scale is stuck.

BBB Error! There is not a single light or heavy ball (or scale is stuck).

BBL Ball #12 is light

BBR Ball #12 is heavy

BLB Ball #11 is light

BLL Ball #9 is heavy

BLR Ball #10 is light

BRB Ball #11 is heavy

BRL Ball #10 is heavy

BRR Ball #9 is light

LBB Ball #8 is light

LBL Ball #6 is light

LBR Ball #7 is light

LLL Error! Scale is stuck!

LLB Ball #2 is heavy

LLR Ball #1 is heavy

LRB Ball #5 is light

LRL Ball #3 is heavy

LRR Ball #4 is heavy

RBB Ball #8 is heavy

RBL Ball #7 is heavy

RBR Ball #6 is heavy

RLB Ball #5 is heavy

RLL Ball #4 is light

RLR Ball #3 is light

RRB Ball #2 is light

RRL Ball #1 is light

RRR Error! Scale is stuck!

We know of two possible solutions:

**Solution 1**Let's mark the balls using numbers from 1 to 12 and these special symbols:

x? means I know nothing about ball number x;

xL means that this ball is maybe lighter then the others;

xH means that this ball is maybe heavier then the others;

x. means this ball is "normal".

At first, I lay on the left pan balls 1? 2? 3? 4? and on the right pan balls 5? 6? 7? 8?.

If there is equilibrium, then the wrong ball is among balls 9-12. I put 1. 2. 3. on the left and 9? 10? 11? on the right pan.

If there is equilibrium, then the wrong ball is number 12 and comparing it with another ball I find out if it is heavier or lighter.

If the left pan is heavier, I know that 12 is normal and 9L 10L 11L. I weigh 9L and 10L.

If they are the same weight, then ball 11 is lighter then all other balls.

If they are not the same weight, then the lighter ball is the one up.

If the right pan is heavier, then 9H 10H and 11H and the procedure is similar to the former text.

If the left pan is heavier, then 1H 2H 3H 4H, 5L 6L 7L 8L and 9. 10. 11. 12. Now I lay on the left pan 1H 2H 3H 5L and on the right pan 4H 9. 10. 11.

If there is equilibrium, then the suspicious balls are 6L 7L and 8L. Identifying the wrong one is similar to the former case of 9L 10L 11L

If the left pan is lighter, then the wrong ball can be 5L or 4H. I compare for instance 1. and 4H. If they weigh the same, then ball 5 is lighter the all the others. Otherwise ball 4 is heavier (is down).

If the left pan is heavier, then all balls are normal except for 1H 2H and 3H. Identifying the wrong ball among 3 balls was described earlier.

**Solution 2**This solution was provided by Charles Naumann. His method also solves it with just three weighings:

Label the balls 1-12

First Weighing:

Left: 1 2 3 4

Right: 5 6 7 8

Off: 9 10 11 12

Record the heavier side (L, R, or B)

Second Weighing:

Left: 1 2 5 9

Right: 3 4 10 11

Off: 6 7 8 12

Record the heavier side (L, R or B)

Third Weighing:

Left: 3 7 9 10

Right: 1 4 6 12

Off: 2 5 8 11

Record the heavier side (L, R, B)

There are 27 (3^3) possible combination of scale readings. A complete sorted list of the scale reading appears below. Note that only 24 of the 27 readings should be possible given the original problem statement. The algorithm was designed so that if all three scale readings are the same, an error is flagged indicating that the scale is stuck.

BBB Error! There is not a single light or heavy ball (or scale is stuck).

BBL Ball #12 is light

BBR Ball #12 is heavy

BLB Ball #11 is light

BLL Ball #9 is heavy

BLR Ball #10 is light

BRB Ball #11 is heavy

BRL Ball #10 is heavy

BRR Ball #9 is light

LBB Ball #8 is light

LBL Ball #6 is light

LBR Ball #7 is light

LLL Error! Scale is stuck!

LLB Ball #2 is heavy

LLR Ball #1 is heavy

LRB Ball #5 is light

LRL Ball #3 is heavy

LRR Ball #4 is heavy

RBB Ball #8 is heavy

RBL Ball #7 is heavy

RBR Ball #6 is heavy

RLB Ball #5 is heavy

RLL Ball #4 is light

RLR Ball #3 is light

RRB Ball #2 is light

RRL Ball #1 is light

RRR Error! Scale is stuck!

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