Can you solve it? Alphabet soup

Today, for a change, a word puzzle. Place a different letter in each of the 26 empty white cells of the grid below to make ten common English words. Each letter of the alphabet is used exactly once. The words read along the horizontal lines.

PLEASE NO SPOILERS

I’ll be back with the solution at 5pm UK.

Last column, I set the problem below, which created a lot of interesting discussion. I’m revisiting it now to show a particularly ingenious solution sent in by a reader.

The three directors of a bank are deeply suspicious of one another, and agree a system of locks and keys for the bank’s safe, such that:

What is the smallest number of locks and keys they need to open the safe, and how do they distribute them?

My answer was three locks and six keys (two per lock). If the locks are A, B and C, then the keys need to be distributed as follows: A and B to one director, B and C to another, and A and C to the third. In this way, no single director can open the safe, but any combination of two directors can.

However, there is a better solution! If one allows the locks to be connected to flexible chains (and why not), it is possible to meet the demands of the question with only three locks and three keys, a solution sent in by Mate Puljiz. He drew a picture of his solution below, saying that one director is given the key for A, one the key for B and one the key for C.

If you unlock B, the A-chain is clearly still interlocked with the C-chain. If you unlock C, the A-chain remains interlocked with the B-chain. And if you unlock the A-chain, you get the situation below (with B shortened to make the loops easier to see), in which the other two are still interlocked. Thus no director can open the safe on their own. But if any two locks are unlocked, the third lock will unthread itself from the handle and the safe will open. Neat!

Puljiz’s solution links to the maths of the Borromean rings. If you would like to explore this area further, a good start is this article about picture-hanging puzzles.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

I’m the author of several books of popular maths and puzzles. The most recent, So You Think You’ve Got Problems is just out in paperback. You can get buy it here at The Guardian Bookstore.

If you are reading this in the Guardian app, and you want a notification each time I post a puzzle, or its solution, click the ‘Follow Alex Bellos’ button above.

Thanks to Shivaja Prabod for sending in this week’s puzzle. He remembers seeing it a decade ago and writing it down, but he can’t remember the original source.