Puzzle 8 A Casual Chat.pdf

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Usopp, Franky, and Chopper are hanging out when Chopper suddenly gets an idea. “Guys, pick a random whole number from 1 to 3000,” he tells Usopp and Franky. “But don’t tell each other — just tell me.”

They do as he says, and afterward, Chopper announces, “One of your numbers is exactly 7 times the other. Do you know the other person’s number?”

Usopp thinks for a second. “Nope.”

Afterward, Franky tilts his head. “Hmm… Nah, I don’t know either.”

Robin, who has been quietly listening from afar, smiles.

Now I know Franky’s number.

How does she know?

[Note: Chopper, Franky and Usopp are smart]

Answer

Franky's number is 49

Explanation

This type of puzzle is very interesting as it seems impossible to solve at first. However, once you know the trick, they are still very tricky to solve. 😅 This one is on the easier side compare to its peers.

Call Usopp's number U and Franky's number F. From what Chopper said, we know that U and F are somewhere between 1 and 3000, and U x 7 = F or F x 7 = U

So, 0 ≤ U,F ≤ 3000 and 7U = F or 7F = U

Let's look at Usopp's statement: "Nope"

If U was just a random number, say 8. 8 is not divisible by 7, thus Usopp would've immediately known that F = 8 x 7 = 56 and said "Yes." In reality, he said "Nope"

=> U must be divisible by 7.

The idea is that for someone to have said "No," they must've been unsure about whether to multiply or divide their number by 7.

Franky, being a smart man, could also deduce that U is divisible by 7. With the same reasoning, we know that F is also divisible by 7.

If F=7k (k being whole number not divisible by 7), Franky would've known that U=7 x 7k because U can't be F ÷ 7=k -- that would mean U=k and U is not divisible by 7, contradicting what we've deduced.

For example, k=4 and F=28

Franky knows his number is F=28 => Usopp number must be U=28÷7 = 4 OR U=28x7=196.

If U=4 then Usopp should've said "Yes, I know Franky's number" as 4 is not divisible by 7 while 4x7=28. However, Usopp said "No," so U cannot be 4

Thus, Franky would've known for sure that U=196 and said "Yes". But, he didn't

=> This scenario is wrong

=> F cannot be 7k

=> F needs to be a number that once divided by 7 is still divisible by 7

=> F = 7*7k = 49k OR F = 7*7*7k = 343k

=> F = 7*7k = 49k OR F = 7*7*7k = 343k

(if F= (7^4)k = 2401k, then for any value of k, F would be too big to multiply by 7 (exceeding 3000)

=> Franky would've known he had to divide by 7 => He would've said "Yes")

If F = (7^3)k = 343k, Franky would've known that U = F ÷ 7 because if U = 7F = 7 x 343k = 2401k then for any value of k, U would be too big to multiply by 7 (exceeding 3000) => Usopp would've known he had to divide by 7 => He would've said "Yes."

For example, k=1 and F=343

Franky knows his number is F=343 => Usopp number must be U=343÷7 = 49 OR U=343x7=2401.

If U=2401 then Usopp should've said "Yes, I know Franky's number" as 2401x7>3000 while 2401÷7=343. However, Usopp said "No," so U cannot be 2401

Thus, Franky would've known for sure that U=49 and said "Yes". But, he didn't

=> This scenario is wrong

=> F cannot be 343k => F needs to be a number that once multiplied by 7 can still be multiplied by 7 a second time and not exceed 3000

=> F = 7*7k = 49k

Thus, F= 49k.

If k≥2, then F ≥ 2 x 49k => F ≥ 98k. If U = 7F, then U ≥ 7 x 98k = 686k. Again, for any value of k, 686k is too big to multiply by 7 (exceeding 3000) => Usopp would've known he had to divide by 7 => He would've said "Yes."

For example, k=2 and F=98

Franky knows his number is F=98 => Usopp number must be U=98÷7 = 14 OR U=98x7=686.

If U=686 then Usopp should've said "Yes, I know Franky's number" as 686x7>3000 while 686÷7=98. However, Usopp said "No," so U cannot be 2401

Thus, Franky would've known for sure that U=14 and said "Yes". But, he didn't

=> This scenario is wrong

=> F cannot be 49k with k≥2

However, when k=1, U = 343, U is small enough to be multiplied by 7, leaving Usopp unsure whether to multiply or divide by 7.

Therefore, k=1 and F = 49. Usopp's number is either 7 or 343

And that's how Robin knows Franky's number. Pretty cool, right?

Free talk

Took me quite a bit of time to censor the picture, but I think I did fine. To be honest, it was probably uncecessary - the original is like 13+. https://onepiece.fandom.com/wiki/Tony_Tony_Chopper/Personality_and_Relationships?file=Straw_Hats_Goof_Off_at_Reunion.png

Tell me if you notice that it has been edited