Varsity Math, Week 152

Varsity Math 152

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Welcome to Varsity Math, the weekly math puzzle column by the National Museum of Mathematics and featured each weekend in the Wall Street Journal.

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Nuts and Bolts

A customer at the hardware store has purchased a package of five nuts and one of eight bolts, each package priced separately. If the bill for the nuts and the bill for the bolts are multiplied or added, the result is $7.20 either way.

What are the prices of nuts and bolts individually if neither item costs more than a dollar?

Da Bridge

Bridge Crossing

Six hikers must cross a bridge. A maximum of two people can cross at one time. It is night, and they need their one flashlight to guide them on any crossing. Each person walks at a different speed: Person 1 takes 1 minute to cross; Person 2 takes 3 minutes; Person 3 takes 4 minutes; Person 4 takes 6 minutes; Person 5 takes 8 minutes and Person 6 takes 9 minutes. A pair must walk together at the rate of the slower person’s pace. They all begin on the same side.

What is the least amount of time they need to get all six across the bridge?

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Solutions to week 151

To solve Three Burning Ropes, simultaneously light the first rope from one end and the second rope from both ends, reducing the first rope to 12 minutes remaining when the second rope is consumed. At that instant, burn the third rope from one end and the first rope from both ends, reducing the third rope to 122 minutes remaining when the first rope is consumed. Now, burn the third rope from both ends for a 61-minute interval. In Prime Perimeter, John has 7 and 17 on his forehead.

Prime Perimeter answer explained:
John sees (13, 17) on Laura’s forehead and realizes that the only primes that can be on his forehead come from the set 7, 11, 13, 17, 23, and 29. The possible pairs of these are (7, 17) and (11, 13). John now knows that Laura may be looking at either (7, 17) or (11, 13) and announces “I don’t know my numbers.”

If Laura were looking at (7, 17), she would reason that the only primes that could be on her forehead come from the set 13, 17, 19, and 23. The only pairs that work are (13, 17) and (17, 19). By reasoning identical to John’s above, Laura would know that if she has (13, 17), then John would announce that he doesn’t know his numbers. Laura would further reason that if she has (17, 19), then John would have noted that the only primes that could be on his forehead come from the set 5, 7, 11, 17, 23, and 31. The possible pairs that work from this set are (11, 31) and (23, 31). Thus, if Laura were looking at (7, 17), she would be undecided between (13, 17) and (17, 19) on her forehead and would announce that she doesn’t know her numbers.

If Laura were looking at (11, 13), she would reason that the only primes that could be on her forehead come from the set 5, 7, 13, 17, 19, and 23. The only pairs that work are (5, 13) and (13, 17). Laura would further reason that if John had seen (5, 13), he would have known that the only primes that could be on his forehead are (11, 13) and he would have announced his numbers. Since John didn’t announce his numbers, Laura would know her numbers were (13, 17).

Since Laura doesn’t know her numbers, she cannot be looking at (11, 13) and John must have 7 and 17 on his forehead.

Recent Weeks

Week 151: Three Burning Ropes & Prime Perimeter, solutions to Make 29 & Circle of Friends

Week 150: Make 29 & Circle of Friends, solutions to Tile the Square & Largest Circle

Week 149: Tile the Square & Largest Circle, solutions to Cats and Dogs & Sporting Question

Week 148: Cats and Dogs & Sporting Question, solutions to Alternative Sum and Product & Math Quiz

Week 147: Alternative Sum and Product & Math Quiz, solutions to Mailing Problem & Pool Party

Links to all of the puzzles and solutions are on the Complete Varsity Math page.

Come back next week for answers and more puzzles.