Can you solve it? Are you craftier than a cat burglar?

The Guardian

12-05-2025

Two very different puzzles today. In the first, you have to identify two numbers. In the second, you have to escape off a roof.
1. Go compare!
A dealer places one hundred cards on a table. On their face-down sides are the numbers from 1 to 100. The cards are randomly arranged so you have no idea at the beginning which card is which. Your task is to identify the 1 card and the 100 card without turning any of them over.
The only way to learn information about the cards is by comparison. At any stage, you may choose two and ask the dealer which is smaller and which is larger. The dealer always knows. They will never tell you the number on the cards, just which is smaller and which is larger.
It is possible to identify the 1 card after asking the dealer to make 99 comparisons. First, ask them to compare any two cards. Make a note of the lower card, and ask them to compare it with one of the 98 remaining cards. Make a note of the lower card, and ask them to compare it with one of the 97 remaining cards. And so on. The lower card in the 99th comparison must be lower than all other cards, and thus is the 1 card. Likewise you can identify the 100 card after 99 comparisons, making a total of 198 comparisons to find both highest and lowest cards.
Can you find a method to identify the 1 and the 100 cards using less comparisons? What's the optimal strategy?
2. The rope trick
You are a burglar at the top of a 20m building, which has a ledge half way down on which it is possible to stand. There are hooks at the top of the building and on the ledge. You have a 15m length of rope and a knife. You can cut the rope if you like, and also make any type of knot anywhere on the rope, which uses up no length, and which can be placed on either hook.
How would you use the rope to descend the building safely? You are not allowed to jump off the building or the rope.
I'll be back at 5pm UK. PLEASE NO SPOILERS. Please discuss your favourite knots.
Thanks to Geza Bohus for suggesting today's puzzles. Geza was a Hungarian maths olympiad contestant many moons ago and is now semi-retired after a career in academia and industry, specialising in machine learning and financial modelling. These are two of his favourite puzzles.
I've been setting a puzzle here on alternate Mondays since 2015. I'm always on the look-out for great puzzles. If you would like to suggest one, email me.