1) The most simple of thought processes will unravel this puzzle, but you have to focus. Here’s the situation: you have been kidnapped by a gang that disdains Mensa. The gang seats you at a table in a semi-dark room and empties your pockets. You had 12 coins, which the gang places flat on the table in front of you, arranging that 5 coins are heads up and 7 are tails up. You can see the 12 coins, but can’t make out which coins are heads up and which are tails up. You are given rubber gloves so you can’t distinguish heads from tails by feel. Your task is to separate the coins into 2 piles so that there is an equal number of heads in each pile. You are allowed to move or turn over any coins while you do this, but because of the dark and the gloves, you can’t tell whether a particular coin is heads or tails up. Can you prove the gang wrong, and move/flip coins so that when the lights go on everyone will see 2 groups of coins in which there are an equal number of heads up?
2) An even easier problem demands focus as well. There are ten Court jewelers who each gave the Queen a stack of ten gold coins. The real coins weighed 1 ounce each, but one jeweler provided coins from which 0.1 ounce had been shaved off the edge. The Queen belonged to Mensa, and with just one weighing on a scale which showed the weight of coins on the scale she managed to find which stack was short-weighted and therefore which jeweler had shaved his coins. How did she do this?
The answers to last month’s puzzles were supplied last month.
Here are the answers to this month’s puzzles:
1) Think about it. Separate the coins into a group of 5 and a group of 7. Then flip over the smaller group of coins. You’ll now have the result you want. How did you arrive at this method? Easy. When you separated the coins, you considered that you had x heads in the 7 coin pile. This gives you 5-x heads in the smaller pile. The number of tails in the smaller pile will be its total number of coins minus its number of heads, which is 5 – (5-x) = x tails. By flipping all the coins in the smaller pile, you’ve converted the heads into tails, and the x tails into x heads. But there were x heads in the larger pile which you’ve left untouched. Therefore there are now x heads in each pile, and the gang’s disdain for Mensa is proved unfounded.
2) The Queen labeled the stacks #1, #2, #3,…..#10. She took one coin from stack #1, two coins from stack #2, etc, which gave her 55 coins. If the coins were all correct, the total weight would be 55 ounces. A deficiency of .3 ounces means 3 light coins, taken from stack #3. A deficiency of .5 ounces means 5 light coins, taken from stack #5. Etc. Thus one weighing is all she needed.
[This month’s puzzles are adapted from Posamentier and Lehmann’s Mathematical Amazements and Surprises]