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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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### Paradoxical Averages

Three families are the only occupants of an apartment building. In this building, the average number of children per family is 6. Also in this building the average number of siblings per child is 8.

*How can this happen? Find both possible cases.*

### Average Trial Length

Team members conduct a number of trials by rolling a fair 6-sided die until it shows a 6. If the first roll is a 6 or all prior rolls in the trial are even, they record how many rolls it took; otherwise they discard the trial.

*What is the expected number of rolls per successful trial?*

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## Solutions to week 153

In Twelve Statues, one sibling gets the statues of heights 1, 2, 4, 8, 9, and 12 centimeters. The other gets the statues of heights 3, 5, 6, 7, 10, and 11 centimeters. In Faculty Meeting, the children are 9, 9, and 19.

**Twelve Statues answer explained:**

To get equal weights for the sets, we need the sum of the cubes of the heights to be the same for the two sets of statues. This is uniquely accomplished with 1 + 8 + 64 + 512 + 729 + 1728 = 27 + 125 + 216 + 343 + 1000 + 1331.

Accordingly, one sibling gets the statues of heights 1, 2, 4, 8, 9, and 12 centimeters. The other gets the statues of heights 3, 5, 6, 7, 10, and 11 centimeters.

**Faculty Meeting answer explained:**

The children are 9, 9, and 19 so Sum = 37 and Sum of Cubes = 8317. Smith guessed (5, 16, 16) for the ages. Two years ago, the ages were (7, 7, 17) and Jones guessed (3, 13, 15).

## Recent Weeks

**Week 153**: Twelve Statues & Faculty Meeting, solutions to Nuts and Bolts & Bridge Crossing

**Week 152**: Nuts and Bolts & Bridge Crossing, solutions to Three Burning Ropes & Prime Perimeter

**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

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

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