Brooklyn-99 Puzzle

Not sure if anyone here watches Brooklyn 99, but there was a puzzle a few episodes ago and I just wasted a bunch of time figuring out a solution, so I’m gonna clog up this blog by posting it here.  Enjoy!

Puzzle: You have 12 islanders, and one of them is a different weight.  You have a seesaw to weight them with (i.e. it will only tell if a batch of islanders is heavier or lighter than another batch) but may only use it three times.  How do you figure out which islander is different?

My Solution, in the form of solving for each individual islander:

HEAVY and LIGHT are interchangeable (i.e. if solving for opposite, simply reverse all instances of ‘HEAVY’ with ‘LIGHT’ and vice versa)

First four are relatively straightforward, can be done by process of elimination

1. Weighings that will establish 1 as HEAVY:
5-6-7-8 vs 9-10-11-12 : SAME
now you know that any ball from 5-12 is identical and that target is 1, 2, 3, or 4
1-2 vs 3-5 : 1-2 is HEAVIER
here you have established that 1 or 2 is a heavy ball or 3 is a light one
1-3 vs 5-6 : 1-3 is HEAVIER
1 = HEAVY

2. Weighings that will establish 2 as HEAVY:
5-6-7-8 vs 9-10-11-12 : SAME
1-2 vs 3-5 : 1-2 is HEAVIER
again, here you have established that 1 or 2 is a heavy ball or 3 is a light one
1-3 vs 5-6 : SAME
remember you have established that 1 or 2 is a heavy ball, and since 1 is not it…
2 = HEAVY

3. Weighings that will establish 3 as LIGHT:
5-6-7-8 vs 9-10-11-12 : SAME
1-2 vs 3-5 : 1-2 is HEAVIER
again, here you have established that 1 or 2 is a heavy ball or 3 is a light one
1-3 vs 5-6 : 1-3 is LIGHTER
3 = LIGHT

4. Weighings that will establish 4 as HEAVY:
5-6-7-8 vs 9-10-11-12 : SAME
1-2 vs 3-5 : SAME
4 vs 5 : 4 is HEAVIER
4 = HEAVY

5. Weighings that will establish 5 as HEAVY:
5-6-7-8 vs 9-10-11-12 : 5-6-7-8 is HEAVIER
here you have established that either 5-6-7-8 is contains a heavy ball, or 9-10-11-12 contains a light one
5-6-9 vs 7-8-11 : 5-6-9 is HEAVIER
here you have established that either 5 or 6 is a heavy ball, or that 11 is a light one
6-11 vs 1-2 : SAME
5 is HEAVY

6. Weighings that will establish 6 as HEAVY:
Note that 6 is same as 5 except for last weighing
5-6-7-8 vs 9-10-11-12 : 5-6-7-8 is HEAVIER
5-6-9 vs 7-8-11 : 5-6-9 is HEAVIER
6-11 vs 1-2 : 6-11 is HEAVIER
6 is HEAVY

7. Weighings that will establish 7 as HEAVY:
5-6-7-8 vs 9-10-11-12 : 5-6-7-8 is HEAVIER
5-6-9 vs 7-8-11 : 5-6-9 is LIGHTER
here you have established that either 9 is a light ball or that 7 or 8 is a heavy one
8-9 vs 1-2 : SAME
7 is HEAVY

8. Weighings that will establish 8 as HEAVY:
Note that 8 is same as 7 except for last weighing
5-6-7-8 vs 9-10-11-12 : 5-6-7-8 is HEAVIER
5-6-9 vs 7-8-11 : 5-6-9 is LIGHTER
8-9 vs 1-2 : 8-9 is HEAVIER
8 is HEAVY

9. Weighings that will establish 9 as LIGHT:
Note that 9 is same as 7 except for last weighing
5-6-7-8 vs 9-10-11-12 : 5-6-7-8 is HEAVIER
5-6-9 vs 7-8-11 : 5-6-9 is LIGHTER
8-9 vs 1-2 : 8-9 is LIGHTER
9 is LIGHT

10. Weighings that will establish 10 as LIGHT:
5-6-7-8 vs 9-10-11-12 : 5-6-7-8 is HEAVIER
5-6-9 vs 7-8-11 : SAME
10 vs 1 : 10 is LIGHTER
10 is LIGHT

11. Weighings that will establish 11 as LIGHT:
Note that 11 is same as 5 except for last weighing
5-6-7-8 vs 9-10-11-12 : 5-6-7-8 is HEAVIER
5-6-9 vs 7-8-11 : 5-6-9 is HEAVIER
6-11 vs 1-2 : 6-11 is LIGHTER
11 is LIGHT

12. Weighings that will establish 12 as LIGHT:
5-6-7-8 vs 9-10-11-12 : 5-6-7-8 is HEAVIER
5-6-9 vs 7-8-11 : SAME
10 vs 1 : SAME
12 is LIGHT