# Dot Product Riddle

1. Feb 10, 2013

### Von Neumann

I was recently posed a riddle that went like the following:

There are two people. Person A picks three numbers from 0-99. Person B guesses which three numbers that person A has picked. Then, person A gives the dot product of his picked numbers with person B's guessed numbers. The question is how could person B figure out person A's selected numbers in three guesses. Even more challenging is to provide a solution that allows person B to guess the numbers in one guess.

I have a solution to the first part:

Think of person A and person B as having their guess put into vectors $\vec{a}$=(a$_{1}$, a$_{2}$, a$_{3}$) and $\vec{b}$=(b$_{1}$, b$_{2}$, b$_{3}$) respectively. To get the corresponding component a$_{1}$, person B should select the components (1,0,0) so the dot product will yield a$_{1}$. Same for a$_{2}$ and a$_{3}$. Simple enough.

The next part I am stumped. The only clue I was given is that person B's three guesses are not restricted between 0-99. Anyone have any insight?

2. Feb 10, 2013

### Ferramentarius

The key is that the numbers have a finite length, and can be separated far enough from each other by multiplication for further examination.

3. Feb 10, 2013

### Von Neumann

Hey guys!

I had no idea it was so simple! I was looking into it too much. Thanks for the help.

4. Feb 15, 2013

This also depends upon the value incorporated --In case one is choosing (7 ,77 ,93) as the three value -it may lead to come with 3 guesses .
Finite length numbers can be taken as simple guesses.Comes handy only with vectors

5. Feb 15, 2013

### autodidude

I'm probably way off here but don't you get a single equation with 3 unknowns? (and a restricted domain)

6. Feb 15, 2013

### D H

Staff Emeritus
That's one additional guess, not one guess. The conversation would go like this:

Person A: I've picked three numbers from 0-99. Can you guess what they are, in the order in which I picked them? As a hint, I'll tell you the inner product of my numbers and your guess if your guess is wrong.
Person B: OK. Here's my first guess: b1, b2, and b3.
Person A: Hey! That's cheating! It's also wrong. But since I didn't make my rules clear enough, I guess I'll have to tell you that the inner product is c.
Person B: OK! Here's my second guess: a1, a2, and a3.
Person A: Correct.

Two guesses, not one. The puzzle is how to frame the first guess so that the second guess will inevitably be correct.

No. There is a way (there are an infinite number of ways) to formulate the initial guess so that the second guess will always be correct.

7. Feb 15, 2013

### autodidude

How?

8. Feb 16, 2013

### Mute

Read Ferramentarius' clue again, and note that the guesses for B are not restricted to the range 0-99.