Varsity Math, Week 151

Varsity Math 151

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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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Rope

Three Burning Ropes

You have three ropes and some matches. The ropes burn irregularly like fuses when lit at either end. The first rope burns in 48 minutes, the second rope burns in 72 minutes, and the third rope burns in 128 minutes.

How can you produce a time interval of 61 min?

Prime Perimeter

Coach: “John and Laura, I have written two prime numbers, possibly the same, on each of your foreheads. You can only see what I’ve written on the other person and you don’t both have the same pair. Any three of the four numbers I’ve written form the sides of a triangle with a perimeter that is a prime number.”

John (who sees 13 and 17 on Laura’s forehead): “I don’t know my numbers.”

Laura, after hearing John’s statement: “I don’t know my numbers.”

What numbers are on John’s forehead?

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Solutions to week 150

In Make 29, 29 = (6 – .2) ÷ .2. In Circle of Friends, there are 18 members in the circle.

Make 29 answer explained:
29 = 58/2 = 5.8/.2 = (6-.2) ÷ .2

Circle of Friends answer explained:
With some analysis you will find that there is no solution if the number of members is a multiple of four. Each other number has a unique solution as follows.

5: (1, -3, 3, 9, 5)
6: (4, -1, 0, 7, 8, 3)
7: (8,2, -4, 4, 12, 6, 0)
9: (1, -7, 3, 13, 5, -3, 7, 17, 9)
10: (6, -3, -2, 9, 10, 1, 2, 13, 14, 5)
11: (12, 2, -8, 4, 16, 6, -4, 8, 20, 10 ,0)
13: (1, -11, 3, 17, 5, -7, 7, 21, 9, -3, 11, 25, 13)
14: (8, -5, -4, 11, 12, -1, 0, 15, 16, 3, 4, 19, 20, 7)
15: (16, 2, -12, 4, 20, 6, -8, 8, 24, 10, -4, 12, 28, 14, 0)
17: (1, -15, 3, 21, 5, -11, 7, 25, 9, -7, 11, 29, 13, -3, 15, 33, 17)
18: (10, -7, -6, 13, 14, -3, -2, 17, 18, 1, 2, 21, 22, 5, 6, 25, 26, 9)

The next case where Cindy could have made her statement involves a circle of 90 friends where she announces 5 and was initially given number 50. The next two cases after that are a circle of 162 friends where she announces 9 and a circle of 234 friends where she announces 13.

Recent Weeks

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

Week 146: Mailing Problem & Pool Party, solutions to Find m and n & Escape Room

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

Come back next week for answers and more puzzles.