## ________________

A bit of geometry is what’s challenging the team this week.

## ________________

### Mailing Problem

Back from a foreign trip, the coach reports an interesting pricing policy at a certain country’s post office. A package is charged according to the total of length + width + height of the package, where the three dimensions are whole numbers. The coach had considered a box with integers for each of the dimensions but found by changing each dimension by no more than one unit, the sum could be reduced while retaining the same volume.

*If the volume of the coach’s package was between 300 and 400, what were the dimensions of the package?*

### Pool Party

The coach has invited the team to a pool party at his house. The pool is circular and the largest possible for the coach’s yard. It just touches the corner of a 2 foot by 9 foot walkway nestled in the lower left corner of the yard as shown.

*How big is the coach’s yard?*

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## Solutions to week 145

In Find m and n, m = ± 59 and n = 6. In Escape Room, there are seven people in the larger group and the ratio of the two possible total charges is two.

**Find m and n answer explained:**

Rewrite the equation as 615 = 4^{n} – m^{2}. Then 615 = 3 × 5 × 41 = (2^{n} – m) × (2^{n} + m). Note that the sum of the two factors of 615 must be 2^{n+1} and the only factors giving this are 5 and 123. This produces m = ±59 and n = 6 as the only answers.

**Escape Room answer explained:**

Let G = the number of people in the larger group, P = the initial price per minute, and T = the time limit. If the larger group spent a time, t_{1}, between T/2 and T, then their total charge was PGt_{1} and the total charge for the group of three was 3P[T + (2t_{1} – T) × 3/2]. If these are equal, then t_{1} = 1.5T/(9 – G) and the total charge for the two groups is C_{1} = 3GPT/(9 – G).

## Recent Weeks

**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

**Week 142**: A Golden Set & Combining Rule, solutions to Sudoku Variant & Making 8 and 16

**Week 141**: Sudoku Variant & Making 8 and 16, solutions to Nick’s Birthday & Burning Ropes

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

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