Gauge her age: The Weekly Puzzle by Dilip D’Souza

A brain-teaser to set your neurons firing. This one involves a neat magic trick you can perform using just numbers.

Namita and Taran have set up a blind date. Now, Taran is a well-brought-up young man, and knows he must not ask a lady her age. But he’s also an intensely curious young man, and he thinks he has a way to find out.

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So while they are waiting for their masala dosas at Balaji Restaurant, Taran asks Namita to think of any positive integer (1, 2, 56, 22708, etc), but not to tell him what it is.

“Multiply it by 8,” he says. She purses her lips for a couple of seconds, then nods her head. “Now add your age in completed years.” She nods again, a small smile playing on her lips. “Finally, add your original number and tell me what you have.”

Here’s the thing: Within a few seconds of Namita telling him her total, Taran will guess Namita’s age, confident that it will be right. In the small chance that he gets it wrong, he will take a second guess and that one will almost certainly be right.

If he’s still wrong, he’ll admit defeat.

But he guesses right the first time. Namita is totally charmed. “The things that numbers can do for you,” says Taran. Shyly, they hold hands …

… but wait, how does this work? (It really does).

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How to solve it?

This has to do with two things — digital roots and the number 9.

Given any integer (1, 2, 13, 56, 22708, etc), add its digits and keep adding till you get a single digit. That’s its digital root (DR). Thus the DR of 1 is 1. Of 13? 1+3 = 4. 56? 5+6 = 11. 22708? 2+2+7+0+8 = 19.

Also, the DR of 9 or any multiple is 9 itself.

And here’s the heart of it all: if you add a multiple of 9 and any other number, the DRs of the sum and of the original will be the same; eg, the DR of 13 is 4. DR of 36 (multiple of 9) + 13 = DR of 49, which is 4 again.

And that is just what Taran got Namita to do: add her age to a multiple of 9. (If her original number was 3, she multiplied by 8 to get 24, added her age and then 3 again — to get her age + 27, and 27 is a multiple of 9).

So when Namita tells Taran her total, he quickly calculates its DR. He knows that this is also the DR of her age. He starts with this number and keeps adding 9, until he gets to a number that looks about right for Namita. That’s his guess.

Let’s say Namita gives him 134. He adds up the digits to get 8. Looking deep into her eyes, he thinks: “8, 17, 26… aha! She looks like she’s 26!” and that’s what he guesses. If he’s wrong, he’s most likely wrong by 9 years — she’s either 17 or 35, and his next guess will be one of those.

So now you know. Shyly, they hold hands…

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