## ________________

This week the coach has a logic problem and a time problem.

## ________________

### Liars’ Club

Four people meet and make the following statements.

Person 1: One or more of us are lying.

Person 2: Two or more of us are lying.

Person 3: Three or more of us are lying.

Person 4: All of us are lying.

*Which ones are lying?*

### Palindrome Times

A 24-hour digital clock occasionally shows times that are palindromes such as 0:34:30, 21:33:12, and 1:20:21, if we ignore the colons.

*What is the smallest interval between two such times?*

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## Solutions to week 143

In Staircase Numbers, A = 10,203,040,506,070,809 = 1,202,337,603 × 8,486,003 and B = 90,807,060,504,030,201 = 995,465,061 × 91,220,741. In Arithmetical Question, the number of seconds in 6 weeks is exactly the same as 10 factorial.

**Staircase Numbers answer explained:**

The prime factorization of A is 3^{2} × 83 × 102241 × 133,593,067. The only way to get a 10-digit factor is from 9 × 133,593,067 = 1,202,337,603, giving A = 1,202,337,603 × 8,486,003.

The prime factorization of B is 3^{2} × 229 × 547 × 883 × 9,277 × 9,833. The 8-digit and 9-digit factors must both begin with 9 and the 8-digit factor is given by 9122 × 9833 = 91220741, so that B = 995,465,061 × 91,220,741.

**Arithmetical Question answer explained:**

The number of seconds in 6 weeks is 6 × 7 × 24 × 60 × 60 = 3,628,800 = 10 factorial.

## Recent Weeks

**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

**Week 139**: Pondering Productivity & Cornfield Planning, solutions to Close to a Quart & Math Party

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

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