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Counterfeit Checker -- SOLUTION

Posted By: Nack Ballard
Date: Sunday, 17 February 2019, at 9:37 p.m.

In Response To: Counterfeit Checker (Nack Ballard)

Congratulations to Bob Koca, Scott Johnston, Michael Neilsen, Sean Garber, Ian Dunstan, Joe Russell and Leo Bueno, in order of correct submission.

As you might have suspected, Aladdin was wily enough to work out the solution before accepting King Shahryar's proposition.

SOLUTION

Let's number the checkers/coins X, 2, 3, 4, 5, 6, and "v" stands for "versus" (or the balance point of the scale). Textually, "E" means the two sides of the scale are Equal, "H" means the left side is Heavy, and "L" means the left side is Light.

Start with...

....X2 v 34

(a) If E, then X v 5.
.....If E again, the answer is 6; if not then 5. Or

(b) If H or L, then 3 v 4.

Explanation: For (b), if the H/L result changes, the answer is 3 (the moved checker). If the H/L result stays the same, the answer is 4 (the fixed checker). If E, the answer is 2 (the checker taken off the scale).


Alternate method for (b): X5 v 23.

Explanation: For this method, if the H/L result changes, the answer is 3 (the moved checker). If the H/L result stays the same, the answer is 2 (the fixed checker). If E, the answer is 4 (the checker taken off the scale).


Overview: For both "b" methods, the key is to take one of 2, 3 or 4 off the scale and either pair or separate the other two (whichever is a change from the first weigh), thereby diversifying all three results (H, L or E). The distinction of H or L is whether it matches (or not) the H or L in the first weigh.

After the first weigh, X will no longer be your only known-neutral checker. For the second weigh, you can replace X with one of the other neutrals, if you like (makes no difference). Indeed, one must use 5 and/or 6 as a second neutral if the second method is chosen.

(All successful respondents incorporated the first method; except Scott and Joe, who employed the second method.)

If the counterfeit is 6, Aladdin will not know if it is heavy or light, but I doubt this will cause him to lose sleep.

Aladdin made another calculation: He knew his wife, all the more so on Valentine's Day, would have mixed feelings about her future having been "wagered," but as she would no longer be poor either way she would forgive her clever husband.

Aladdin was right.

Nack

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