Exactly who lies? The Weekly Puzzle by Dilip D’Souza
Here’s a riddle to tease your brain. It’s a party, but not everyone deserves the cake and ice-cream.Updated: Oct 18, 2020, 13:44 IST
Roohi is graduating from college and throwing a party. Her 100 friends — yes, exactly 100 — plan a game for her with a twist.
Because all of them (and Roohi) took a logic class that everyone loved, this is a game of logic. It goes like this: In a large conference room they have hired, the 100 friends line themselves along the walls. Roohi will enter, turn left and walk slowly clockwise around the room, listening to each friend in turn. She knows what they say will be identical except for a number.
The first, Romeo, announces: “Exactly one of us will lie to you.” The second, Rati, says: “Exactly two of us will lie to you.” The third is Ramya, and she says: “Exactly three of us will lie to you.” So it goes around the room. The 99th friend is Ronaldo, who says: “Exactly 99 of us will lie to you.” And of course the 100th and last, Rakshanda, says: “Exactly 100 of us will lie to you.”
Roohi’s task is to offer a slice of cake to each friend who tells her the truth.
Questions: Does anyone get cake? If so, how many, and what’s the serial number of the first friend to get cake?
After a few drinks, they play a variation. The friends line up the same way and Roohi starts around the room the same way. Only this time, instead of “Exactly”, each uses “At least”. This time, she must give a scoop of ice-cream to each friend who lies.
Questions: Does anyone get ice-cream? If so, how many, and what’s the serial number of the first friend to get ice-cream?
Scroll down for the solution
Weekly Puzzle Solution
The first time: if a given statement is true, all the others must be lies. (After all, if it’s true that exactly 23 will lie, it cannot also be true that exactly 51 will lie). Thus only one is true and the other 99 are lies, which means that #99, Ronaldo, gets cake. Nobody else.
The second time: Do this with just six friends, it’s easier to visualise. Consider — if a given friend (after #2) tells the truth, the one before her must be telling the truth too (if it’s true that at least 2 will lie, it’s also true that at least 1 will lie). Similarly, if a given friend (before #6) lies, the one after him must be lying as well (if it’s false that at least 4 will lie, it’s also false that at least 5 will lie). So we have to find a number between 1 and 6, call it ‘p’, such that the statements below and up to p are true, and the ones above it (from serial number p+1 up) are lies.
Now of the true pronouncements, the one numbered p actually includes all those before it — and it says at least p statements are lies. And the next statement, at serial number p+1, is a lie, and it says at least p+1 statements are lies. Thus we know at most p statements are lies. So if at least p are lies and at most p are lies, we know exactly p statements are lies. But we also know that the first p statements are true. So of 6 statements, p are true and p are false — thus p must be 3.
Apply this same reasoning to 100 friends and we find that the first 50 tell the truth, the second 50 lie. Thus friend #51, one Rambo, is the first to lie, thus the first to get ice-cream, which Roohi serves to all those after Rambo as well.