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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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### Tile the Square

*Cover a square with four tiles that are isosceles triangles, no two of which are identical. No overlap or overhang of the tiles is allowed.*

### Largest Circle

What is the radius of the largest circle that can be drawn inside the polygon shown below?

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## Solutions to week 148

In Cats and Dogs, the woman is 37 and she must own at least five pets. In Sporting Question, Sportsville has a population of 104 and the team roster has 39 people in spring and 40 people on the Fourth of July.

**Cats and Dogs answer explained:**

The number 57,165 has prime factors of 3, 5, 37, and 103. Possible triples that multiply to 57,165 are (1; 1; 57,165), (1; 3; 19,055), (1; 5; 11,433), (1; 37; 1,545), (1; 103; 555), (3; 5; 3,811), (3; 37; 515), (3; 103; 185), (5; 37; 309), (5; 103; 111), and (15; 37; 103). Only 37 works for the woman’s age. The possible triples are then (1; 37; 1,545), (3; 37; 515), (5; 37; 309), and (15; 37; 103). Since the woman has both cats and dogs, her total number of pets must be four or more and she must have at least five pets as shown by the triple (5; 37; 309).

**Sporting Question answer explained:**

In Sportsville, the community has a population of 104 and the team roster has 39 people in spring and 40 people on the Fourth of July. The number of possible rosters in the spring is N_{s} = 104!/(39! × 65!) and the number of possible rosters on the Fourth of July is N_{4th} = 103!/(40! × 63!). (n! is n factorial and is the product of the integers from 1 to n.) The ratio of these two is N_{s}/N_{4th} = 104 × 40/(65 × 64) = 1.

## Recent Weeks

**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

**Week 145**: Find m and n & Escape Room, solutions to Liar’s Club & Palindrome Times

**Week 144**: Liar’s Club & Palindrome Times, solutions to Staircase Numbers & Arithmetical Question

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

**Come back next week** for answers and more puzzles.