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Counterfeit Checker, Part 2 -- SOLUTION

Posted By: Nack Ballard
Date: Thursday, 28 February 2019, at 2:19 a.m.

Congratulations to Sean Garber, Joe Russell, Bob Koca, and (the determined) Leo Bueno, in order of correct submission.

On the fifth day, Aladdin was able to solve the 15-checker puzzle. He could hardly contain his excitement as he double-checked and triple-checked, and otherwise waited out the final two days for Shahryar to return from overseas.

When the time came, his imperial excellency watched from a high window as a familiar figure approached the castle. Less familiar was the manner of gait: an enthusiastic gallop.

The cagy monarch still pinned his hopes (to claim the two beautiful women his subject would wager against the fifteen gold checkers) on a mistaken answer. However, Aladdin's crisp delivery left no doubt that he had correctly solved the puzzle.

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SOLUTION

"X" is the known-neutral checker, 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.

The first weigh must employ ten checkers, including the X:

..................12345 v 6789X

If it balances, the counterfeit is one of the five checkers not yet weighed. Use the X coin as the sixth and (in the other two weighs) solve as in Part 1.

If it does NOT balance, then take three checkers off the scale, and move three of the (other six) checkers from one side of the scale to the other. (On both counts, I refer to non-X checkers.) To that end, any of the basic scheme examples below work (take your pick). Some use additional X coins (which you can draw from the pool of five coins you set aside, that you now know are non-counterfeit / neutral).

......................167 v 289..............(2,6,7 switch sides), or
....................1267 v 38XX...........(3,6,7 switch), or
..................12367 v 4XXXX........(4,6,7 switch), or
................123678 v XXXXXX.....(6,7,8 switch)

• If the second weigh balances (E), the counterfeit is one of the three checkers taken off the scales.
• If the second weigh is the opposite of the first (H/L result changes), it is one of the three checkers that switched sides.
• If the second weigh agrees with the first (same H/L result), it is one of the three checkers that did NOT switch sides.

Regardless, you have narrowed down the suspect to three checkers, each of which you have potentially identified as heavy or light. Weigh two like ones (one against the other), with the third off the scale. This second diversification of three results (E, H, L) allows you to zero in on the counterfeit.

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Aladdin's brilliant explanation earned him a well-deserved weight to add to his saddle bags. Arguably, though, his presentation was too convincing, making it less likely he will later be offered the same proposition at the four-weigh level.

Then again, even a hoard of 42 gold checker/coins make up an extremely small percentage of King Sharyar's assets.

Nack

Messages In This Thread

• Counterfeit Checker, Part 2 -- SOLUTION
  Nack Ballard -- Thursday, 28 February 2019, at 2:19 a.m.
  • Another solver discovered
    Nack Ballard -- Saturday, 2 March 2019, at 8:50 p.m.