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This past Tuesday, our illustrious math coach was honored as a special guest at the National Museum of Mathematics’ exciting MoMath *Masters* tournament. The students congratulate the coach by posing two puzzles in his honor.

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### Circumnavigation

Imagine you are on an island with two identical airplanes and an unlimited supply of fuel and pilots. Each airplane can go 300 miles on a full tank. Both aircraft fly at the same constant speed, can refuel each other instantly, and consume fuel at the same constant rate. Suppose you are on a planet that is entirely ocean except for the island. You want one airplane to circumnavigate the planet, flying over both the north and south poles before returning safely to the island. The other airplane must safely return to the island as well.

*What is the largest circumference the planet can have and still allow a circumnavigation by one airplane?*

### Typos

A publisher gives the proof sheets of a new book to two different proofreaders. The first reader finds 252 typos; the second finds 260. Strangely enough, only 13 typos are found by both.

*If each typo is independently discovered with equal probability, then how many initial typos are expected to be in the book?*

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## Solutions to week 128

In Keeping Things Even, Brown’s probability of hitting Jones in a single shot is p = 2/3. In Roll Those Bones, your probability of winning is one-half.

**Keeping Things Even answer explained:**

Let p be Brown’s probability of hitting Jones for each shot and let Jones’s probability of winning the duel be

J = .4 + .6(1 – p)J. If we set J = .5 to make the duel fair, then Brown’s probability of hitting Jones in a single shot is p = 2/3.

**Roll Those Bones answer explained:**

The probability of getting no sixes in any one roll is (5/6)^{5} = 3125/7776. The probability of getting one six in any one roll is (5/6)^{4} × (1/6) × 5 = 3125/7776. Since you keep rolling dice until one of these two possibilities occurs, your probability of winning is one-half.

## Recent Weeks

**Week 128**: Keeping Things Even & Roll Those Bones, solutions to Race to the Gate & Tricky Logic Game

**Week 127**: Race to the Gate & Tricky Logic Game, solutions to Logical Coffee Order & Here’s Looking at You

**Week 126**: Logical Coffee Order & Here’s Looking at You, solutions to Piece of Pi & Making 17 and 27

**Week 125**: Piece of Pi & Making 17 and 27, solutions to Cutting the Triangle & Square Repair

**Week 124**: Cutting the Triangle & Square Repair, solutions to Mining Mishap & Pulley Puzzle

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

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