12 Coin Solution
Okay, here's the solution to the
12
coin problem:
Consider the coins are numbered 1-12.
#1: 1-4 vs. 5-8
1.A balance is even:
That means it's in 9-12, but we don't know which one, nor the orientation (heavier or lighter).
#2: 9,10 vs. 1,11 (note that 1 is a known good coin)
2.A even:
That means it's 12
#3: 1 vs. 12
12 tips in direction of orientation
2.B balance tips (either direction):
#3: 9 vs. 10
3.A even:
That means it's the one you took off (11), and from #2 you know the orientation.
3.B balance tips the same way:
That means it's the one that stayed on the same side (9)
3.C balance tips the other way:
That means it's the one that switched sides (10)
1.B balance tips:
#2: 1,2,4 vs. 3,5,9 (note that 9 is a known good coin)
2.A even:
That means it's one you took off (6, 7 or 8)
#3: 6 vs. 7
3.A even:
That means it's the one not put back on (8), orientation known from #1
3.B balance tips the same as #1
That means it's the one that stayed on the same side (7)
3.C balance tips in direction opposite of #1
That means it's the one that switched sides (6)
2.B balance tips same way:
That means it's a coin that stayed on the same side (1,2 or 5)
#3: 1 vs. 2
3.A even:
That means it's the one you took off (5), orientation known from #1 & #2
3.B balance tips same way again:
That means it's the one that never moved (1)
3.C balance tips the other way
That means it's the one that switched sides (2)
2.C balance tips other way:
That means it's a coin that switched sides (3 or 5)
#3: 1 vs. 5
3.A even:
That means it's the one you took off (4), orientation known from #1 or #2
3.B balance tips in the same direction as #2 again:
That means it's the one that was left on (5)
3.C balance tips other way from #2:
That means is's the one that was taken off (3), orientation known from #1 or #2
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