# Puzzles

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## Tags

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Source: Puzzle Critic

Pick a number. Call it \(n\). Write down all the numbers from \(n+1\) to \(2n\) (inclusive). For example, if you picked 7, you would write:

$$8,9,10,11,12,13,14$$
Below each number, write down its largest odd factor. Add these factors up. What is the result? Why?

## Odd Squares

Source: Maths Jam

Prove that 1 and 9 are the only square numbers where all the digits are odd.

## Odd Sums

What is \(\frac{1+3}{5+7}\)?

What is \(\frac{1+3+5}{7+9+11}\)?

What is \(\frac{1+3+5+7}{9+11+13+15}\)?

What is \(\frac{1+3+5+7+9}{11+13+15+17+19}\)?

What is \(\frac{\mathrm{sum\ of\ the\ first\ }n\mathrm{\ odd\ numbers}}{\mathrm{sum\ of\ the\ next\ }n\mathrm{\ odd\ numbers}}\)?