## ________________

A bit of calculation and reasoning is always good exercise for the team.

## ________________

### The Size of Humanity

Two years ago, the Earth’s population reached 7.4 billion people. Consider the Earth as a sphere of radius 6,371 kilometers and spread the volume of humanity over the surface of the Earth in a shell of constant thickness. Assume the average person occupies a volume of 63,000 centimeters^{3}.

*How thick would the shell be?*

### Fish Pond

A fish pond has a horizontal bottom and vertical sides. To make a platform, the owner places three identical, impermeable cubical blocks side by side, resting on the bottom. The first block causes a rise in water level of less than 12 inches. The second and third blocks each cause the level to rise exactly 12 inches.

*How high in inches is the water level above the top of the completed platform?*

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## Solutions to week 129

In Circumnavigation, the circumference can be **no more than 500 miles**. In Typos, the number of expected typos is **5,040**.

**Circumnavigation answer explained:**

Both airplanes fly 100 miles toward one of the poles where airplane 2 tops off airplane 1 and returns to the island as its fuel is consumed. Airplane 1 continues over and beyond that pole, getting to within 100 miles from the island where its fuel is consumed. In the meantime, airplane 2 refuels at the island and takes off at the right time to just meet the incoming airplane 1 as it is running out of fuel. They divide the fuel from airplane 2 to just reach the island as their fuel runs out. The circumference can be no more than 500 miles.

**Typos answer explained:**

Let the total number of typos be T. The first reader finds typos with a probability 252/T and the second reader finds them with a probability of 260/T. We are told that 13 typos were found by both so 13/T = (252/T)(260/T), from which we compute T = 5,040 typos.

## Recent Weeks

**Week 129**: Circumnavigation & Typos, solutions to Keeping Things Even & Roll Those Bones

**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

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

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