The Picture Hanging Puzzle

I will be sharing some interesting maths puzzles here, somewhat semiregularly. If you solve a puzzle – or make some progress – do share your attempt with me. I would love to read your submissions. Interesting and enlightening reader-submitted solutions will be featured along with the following puzzle.

Here’s the first puzzle.

The Problem

You have stolen an exquisite painting from your local art gallery, and given the daring thief that you are, you would like to hang in your living room! You have with you some nails and a long rope.

Level 1

You would like to hang the painting using two nails. To allow for easy removal, you want the painting to fall if you remove any one of the nails from the wall.

Is this feat possible? If yes, then how would you hang the painting? If no, then prove that it is impossible.

Level 2

Now suppose you want to hang the painting using \(n\) nails. You would like to painting to fall on removing any one nail.

For what values of \(n\) is this possible? In the cases it is possible, how would you hang the painting?

Level 3

Suppose you want to hang the painting using \(n\) nails. This time, you would like the painting to fall on removing any \(m\) nails. The painting should not fall if you remove fewer than \(m\) nails. You may assume \(1 \le m \le n\).

For what values of \(m\) and \(n\) is this possible? In the cases it is possible, how would you go about hanging the painting?

Good luck, have fun!

Share your attempts with me!

• If you solved the problem, how did you solve it? Share your solutions, and the best ones will be featured with the next puzzle.

• If you are stuck, share your progress. What have you tried so far? I will give you a hint.

• Let me know if there’s an error or something’s not clear in the puzzle.

• I would love your feedback. Did you find the puzzle interesting? Or was it too easy / too difficult?

• If you have any puzzles that you would like featured in this series, share those with me too!