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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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### Cats and Dogs

A woman of voting age who is under 100 has a number of pets, including both cats and dogs. Her age and street address are both whole numbers. The product of her age, street address and number of pets is 57,165.

*How old is the woman and what is the fewest number of pets she can own?*

### Sporting Question

The town of Sportsville competes twice a year in games with a neighboring town. In the spring, a team roster of n players is drawn at random from the town’s entire population. On the Fourth of July, a team roster of n + 1 players is drawn at random from the population — excluding the mayor, who has important business to which he must attend. The town population has not changed between spring and the Fourth of July, and the number of possible team rosters for the spring and Fourth of July events are also the same.

*What is the size of Sportsville’s roster in the spring and on the Fourth of July if the population of the town is between 20 and 600?*

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## Solutions to week 147

In Alternative Sum and Product, 9 factorial = 1 × 2 × 4 × 4 × 4 × 5 × 7 × 9 × 9. 45 = 1 + 2 + 4 + 4 + 4 + 5 + 7 + 9 + 9. In Math Quiz, Aileen got 5 correct.

**Alternative Sum and Product answer explained:**

This can be solved by writing the prime factorization of 9 factorial = 2^{7} × 3^{4} × 5 × 7 and experimenting to give 9! = 1 × 2 × 4 × 4 × 4 × 5 × 7 × 9 × 9 and 45 = 1 + 2 + 4 + 4 + 4 + 5 + 7 + 9 + 9.

**Math Quiz answer explained:**

Let p be the points possible for each problem; let n be the number of problems; and let letters A, B, L, and S be the points scored by the participants. Aileen’s statement says A + B + L + S = 2np. The other statements give B = (n – 3)p, L = 9p and S = (n – 11)p. It follows from these statements that A + (n – 3)p + 9p + (n – 11)p = 2np. This reduces to A = 5p so Aileen got 5 correct.

## Recent Weeks

**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

**Week 143**: Staircase Numbers & Arithmetical Question, solutions to A Golden Set & Combining Rule

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

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