Ropes, digits and ants

Multiples of 9 have the charming property that when you add their digits, you get another multiple of 9

For a change from the more serious subjects this column usually tackles, here are four simple number delights that, in my experience, never fail to surprise folks who hear them for the first time. I hope you are one of them.

Answers and explanations are at the end. Diving straight in:

Delight #1

Imagine someone lets you into the Wankhede Stadium before a major cricket match. No, no, you’re not out to vandalize the pitch. You merely want to measure the diameter of the stadium. You tie a rope to the fence at some point outside the boundary, walk across the stadium all the way to a point directly opposite, pull the rope tight along the grass, and tie it to a pole there. You note down the length of the rope—100m, say.

Before you head out, you wonder idly: suppose the match begins and we have to lift this rope to allow batsmen to run under it. (Thought experiments allow for funky cricket). Let’s say we have to raise it about 2m at the centre. How much longer will the rope have to be?

a) About 10cm

b) About 1m

c) About 10m

d) About 100m


Delight #2

In a similar vein, let’s say you manage to string a rope tightly around the Earth’s equator—so we know it’s about 40,000km long. Now you want to lift it an even 1m off the surface of the planet, all the way around. How much longer will the rope have to be?

a) About 10km

b) About 6m

c) About 1cm

d) About 500km


Delight #3

With your back turned, I pick a random number (example: 557,180); scramble its digits to get a new number (785,015); subtract the smaller number from the larger (785,015-557,180=227835). Now I cover up one of the digits in the answer at random (7, say) and rattle off the others to you, the budding mathemagician with your back still turned (2, 2, 8, 3, 5). You immediately tell me the number I covered up. How?


Delight #4

You’re a biologist who studies ants. One day, you come up with an idea for an experiment to observe how they move. You take a thousand ants and drop them on a rod that’s 10m long. Each ant immediately begins moving in the direction it’s facing, at a steady 1m every minute. If two ants bump into each other, both turn around and head in the opposite direction. When an ant reaches either end of the rod, it leaps off and continues its journey on terra firma.

What’s the longest you’ll have to wait to be sure the rod is totally free of ants—meaning all have leaped off?

a) A minute

b) An hour

c) 10 hours

d) 10 minutes


Answer #1: About 10cm; actually, about 8cm. Does that astonish you? It seems totally counter-intuitive, but this is a straightforward consequence of the Pythagoras Theorem: the square on the hypoteneuse is the sum of the squares on the other two sides.

Answer #2: about 6m. Astonishing again, counter-intuitive again? This time, we have a straightforward consequence of calculating the circumference of a circle (the Equator) which is 2 x pi x the Earth’s radius. What we’re asking is, how much does the circumference increase by if we increase the radius by 1m? A little arithmetic tells you the answer: 2 x pi, or a little over 6m.

Answer #3: The process of scrambling and subtracting always produces a multiple of 9. (Why that happens, I leave you to work out). Multiples of 9 have the charming property that when you add their digits, you get another multiple of 9. So when I rattle off the digits to you, you add them up (2+2+8+3+5=20) and quickly subtract from the next higher multiple of 9 (27-20=7). There’s your missing digit!

Answer #4: Ten minutes. What’s confusing about this question is what the ants do when they bump heads. They turn around, and maybe they bump again, and they turn around again, and so on. So we’re unsure how to figure out how long they spend on the rod. But this situation is exactly equivalent to assuming that when they meet, they don’t turn around, but instead step to the side and pass each other, continuing in their respective directions. Now we just need to ask what the longest distance is that any ant will travel on that rod before leaping off. That’s clearly 10m (i.e. if you deposited it at one end of the rod and it trudges to the other end). Travelling that far will take it 10 minutes. Ants who start out somewhere along the rod have less than 10m to go, and so will leap off sooner.

So if you go away and return after exactly 10 minutes, you’ll certainly have an ant-free rod. Congratulations.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners.

A Matter of Numbers explores the joy of mathematics, with occasional forays into other sciences.

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