## ________________

The team tackles a number theory puzzle and a reasoning puzzle this week.

## ________________

### Find m and n

Inspired by June 15, the coach has a problem involving the number 615.

*Find all integer solutions for m and n to the equation m ^{2} + 615 = 4^{n}.*

### Escape Room

Jan and two other team members went on an outing to an escape room, in which groups of people find clues to get out of a locked room. In this version, groups are charged by the number of participants as well as by the amount of time spent in the room. Jan says, “We followed a larger group and spent twice as much time in the room as they did, though our total charge was the same as theirs. The charge per person is proportional to the time spent in the room up to a certain time limit. Time spent over that limit is charged at a 50% higher rate. For any given time limit and base price per time, there are two ways the two groups could have had identical charges, depending on whether the larger group spent more or less than the time limit in the escape room.”

*How many people were in the larger group and what is the ratio of the two possible total charges?*

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## Solutions to week 144

In Liar’s Club, Person 3 and Person 4 are both lying. In Palindrome Times, the two closest times are 9:59:59 and 10:00:01, at two seconds apart.

## Recent Weeks

**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

**Week 140**: Nick’s Birthday & Burning Ropes, solutions to Pondering Productivity & Cornfield Planning

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

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