Scene together, taken apart: The Wknd Puzzle by Dilip D’Souza

The sum of two numbers equals the third. You know what the first two are, but you don’t know which represents the sum. Can you guess the third?

Raveena writes three positive numbers (integers, or whole numbers) on pieces of paper. She gives one to each of her three friends, Priyanka, Salman and Diya, asking them not to look at their number. Then she asks them to hold the numbers up against their foreheads, so that each can see the other two numbers but still cannot see their own.

She says: “Each number is different, and two of the three add up to the third. Can you tell what your own number is?”

Priyanka can see 2 on Salman’s forehead and 3 on Diya’s. She says, sadly, “I don’t know what my number is.”

Salman says: “I don’t know either.”

Diya says: “Nope, I too have no clue.”

But now Priyanka says: “Hey! I know my number!”

Question: What was Priyanka’s number?

Bonus: Does Raveena need to tell them the numbers are different?

Hint: Priyanka knows her number has to be one of two possibilities. But only with one of those will both Salman and Diya not know theirs. Why?

Scroll down for the answer.

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Answer:

Priyanka’s number has to be either 1 or 5. Try 1. Salman will see 1 (P) and 3 (D) and think, “Mine has got to be either 2 or 4. Man, I have no idea which.” Diya will see 1 (P) and 2 (S) and think, “Mine has to be either 3 or 1 ... but wait, it can’t be 1 since all three are different! Thus I’ve got 3!” So she will shout out “3!”

But that doesn’t happen. Instead, Diya says she doesn’t know!

Thus Priyanka has 5. You can work through the logic in that case and see that both Salman and Diya won’t know their number.

Bonus: No, she doesn’t need to tell them the numbers are different. That logic, I leave for you to work out.