Puzzle Page

I've had a number of requests to make this solution page, I hope that you haven't consulted it until you've struggled a bit.

Coin Weighing Solution

Break the coins into three groups of four (called A,B and C; I'll call the individual coins a_1, a_2, etc.)
Weigh A vs. B, either they are the same or they are different.

If A=B, then a_1,...a_4, b_1,...,b_4 are the correct weight.
- Weigh c_1 + c_2 vs. c_3 + a_1,
  - if c_1 + c_2 > c_3 + a_1;
    either c_1 > a_1, c_2 > a_1 or c_3 < a_1.
    - Weigh c_1 + c_3 vs. a_1 + a_2,
      - if c_1 + c_3 < a_1 + a_2; c_3 is light.
      - if c_1 + c_3 > a_1 + a_2; c_1 is heavy.
      - if c_1 + c_3 = a_1 + a_2; c_2 is heavy.
  - if c_1 + c_2 < c_3 + a_1;
    either c_1 < a_1, c_2 < a_1 or c_3 > a_1.
    - Weigh c_1 + c_3 vs. a_1 + a_2,
      - if c_1 + c_3 < a_1 + a_2; c_1 is light.
      - if c_1 + c_3 > a_1 + a_2; c_3 is heavy.
      - if c_1 + c_3 = a_1 + a_2; c_2 is light.
  - if c_1 + c_2 = c_3 + a_1;
    either c_4 < a_1 or c_4 > a_1.
    - Weigh c_4 vs a_1 to find out which.
If A is not = B, assume w.l.o.g. that A > B (if not, just change which coins you call A and B), then c_1,...,c_4 are the correct weight.
- Weigh a_1 + a_2 + b_1 vs. a_3 + b_2 + c_1,
  - if a_1 + a_2 + b_1 = a_3 + b_2 + c_1;
    either b_3 < c_1, b_4 < c_1 or a_4 > c_1; (treated as above)
  - if a_1 + a_2 + b_1 > a_3 + b_2 + c_1;
    either a_1 > c_1, a_2 > c_1 or b_2 < c_1; (treated as above)
  - if a_1 + a_2 + b_1 < a_3 + b_2 + c_1;
    either b_1 < c_1 or a_3 > c_1.
    - Weigh b_1 vs. c_1, e.g. to find out which.

Note that regardless of which coin is heavier or lighter, this process always requires three weighings, i.e. we can never find out in less than three. This seems to imply something about the optimality of the strategy. All strategies I tried to construct which could terminate in less than three weighings failed to work for some case.

Integer Puzzle

Oh Domino

No. Here's an easy way to see why. Each whole domino covers both a black and a white square. If you use a broken domino to cover opposite corners, then you've covered two like colored squares with the same domino. It's not possible (without breaking another domino) to cover the remaining squares since the number or black and white squares don't match up.

Pail Puzzle

Rectangle Puzzle

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