Mathematical & Logical Puzzles -A3

9.       The Weights And The Scales

Four weights are quite sufficient for this purpose.  1 gram, 3 grams, 9 grams and 27 grams.

To weight two grams, the 3-gram weight is put on one pan and the 1-gram weight on the other, making a difference of two grams.

To weight five grams, the 9-gram weight is put on one pan and the and the 3-gram weight and the 1-gram weight are put on the other, making a difference of five grams.

By using this method of adding and subtracting weights, the four weights will weigh any number of whole grams from one to forty grams inclusive.

10.     The Efficient Electrician

On the top floor, the electrician shorted five pairs of wires by twisting them together, leaving the last wire free.  Then he walked to the basement and identified the lower ends of the shorted pairs by means of his continuity tester. He labelled the bottom ends of the first shorted pair A1 and A2 respectively, the ends of the second shorted pair B1 and B2 respectively, the ends of the third shorted pair C1 and C2 respectively, the ends of the fourth shorted pair D1 and D2 respectively, the ends of the fifth shorted pair E1 and E2 respectively and the unpaired end he labelled F.  He shorted the bottom ends of the wires by leaving A1 free, connecting A2 to B1, B2 to C1, C2 to D1, D2 to E1, and E2 to F.

Back on the top floor, he removed all the shorts but left the pairs twisted together at insulated portions so that the pairs were still identifiable.  He then checked for continuity between the free wire, which he knew to be the upper end of F and some other wire.  When he found the other wire he was at once able to label it E2 and identify its mate as E1.  He next tested for contintuity between E1 and another end which when found could be marked D2 and its mate D1.  Continuing in this fashion, the remaining ends were easily identified.

11.     How Old Are You?

If when Jones was 28, Smith was 2x years old, then he was x years old when Jones was 3x years old.  Whatever their respective ages, Jones is always the same number of years older than Smith.  Age difference when Smith is x and Jones is 3x is 2x years.

Age difference when Jones is 28 and Smith 2x is 28 - 2x.

Now the age differences are equal, therefore 2x = 28 - 2x  or 4x = 28  or x = 7.

Age difference is 2x or 14 years.  Therefore when Jones was 28 years old, Smith was 14 years old.

Smith is 28 so Jones is now 28 + 14 = 42 years old.

12.     Ann And Mary

Work from the end backwards.  Let Ann be x years old, then Mary was 3x when she was three times as old as Ann.  Age difference is 2x.  Ann will be three times 3x when she is 9x years old.  Mary will be 4½x when she is half as old as 9x.  Ann will be 4½x - 2x or 2½x when Mary is 4½x.  Therefore Mary now being twice as old as 2½x is now 5x.  Ann is 2x years younger or 3x years old.  But we are told that 5x + 3x = 44.  And from this x = 5½.

Mary is 5x or 27½ years old whilst Ann is 3x or 16½ years old.  Easy isn’t it?

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