# Number TheoryWill the balls meet at the same position?

#### evinda

##### Well-known member
MHB Site Helper
Hello!!! At a clock (on which we have the positions $1,2, \dots, 12$) we place at position $1$ a blue ball and at position $2$ a red ball. At discrete times ($1,2,3, \dots$) we shift the two balls. Each time we shift the blue ball by three positions and the red ball by one position. Will the two balls ever meet at the same position?

I have thought the following.

Let $B$ be the position of the blue ball and $R$ the position of the red ball. Then $B=1+3t$ and $R=2+t$, for some $t \in \mathbb{N}$.

$B=R \Rightarrow 1+3t=2+t \Rightarrow 2t=1 \Rightarrow t=\frac{1}{2} \notin \mathbb{N}$.

Thus, the two balls will never meet at the same position. Am I right? #### Klaas van Aarsen

##### MHB Seeker
Staff member
Hello!!! At a clock (on which we have the positions $1,2, \dots, 12$) we place at position $1$ a blue ball and at position $2$ a red ball. At discrete times ($1,2,3, \dots$) we shift the two balls. Each time we shift the blue ball by three positions and the red ball by one position. Will the two balls ever meet at the same position?

I have thought the following.

Let $B$ be the position of the blue ball and $R$ the position of the red ball. Then $B=1+3t$ and $R=2+t$, for some $t \in \mathbb{N}$.

$B=R \Rightarrow 1+3t=2+t \Rightarrow 2t=1 \Rightarrow t=\frac{1}{2} \notin \mathbb{N}$.

Thus, the two balls will never meet at the same position. Am I right? Hey evinda !!

Shouldn't we take into account that for instance position 13 is the same as position 1?
That is, shouldn't it be:
$$B\equiv R \pmod{12}$$ Furthermore, it doesn't say in which direction we shift the balls does it?
We can assume for now that they are shifted in clockwise direction.
But can we later on also check what happens if we arbitrarily allow a shift in either direction? #### evinda

##### Well-known member
MHB Site Helper
Hey evinda !!

Shouldn't we take into account that for instance position 13 is the same as position 1?
That is, shouldn't it be:
$$B\equiv R \pmod{12}$$ Furthermore, it doesn't say in which direction we shift the balls does it?
We can assume for now that they are shifted in clockwise direction.
But can we later on also check what happens if we arbitrarily allow a shift in either direction? $$B\equiv R \pmod{12} \Rightarrow 1+3t \equiv 2+t \pmod{12} \Rightarrow 2t \equiv 1 \pmod{12}$$

And there is no $t$ such that $2t \equiv 1 \pmod{12}$.

Do we also check what happens if we shift all the balls in the other direction? Or also if we shift only for example the blue balls in the other direction? #### Klaas van Aarsen

##### MHB Seeker
Staff member
$$B\equiv R \pmod{12} \Rightarrow 1+3t \equiv 2+t \pmod{12} \Rightarrow 2t \equiv 1 \pmod{12}$$

And there is no $t$ such that $2t \equiv 1 \pmod{12}$.
Exactly. Do we also check what happens if we shift all the balls in the other direction? Or also if we shift only for example the blue balls in the other direction? Suppose we start with the balls at a distance from each other that is odd.
And suppose we do one step, allowing both balls to shift in either direction.
It means there are 4 cases, doesn't it?
What will be the resulting distance? #### evinda

##### Well-known member
MHB Site Helper
Suppose we start with the balls at a distance from each other that is odd.
And suppose we do one step, allowing both balls to shift in either direction.
It means there are 4 cases, doesn't it?
What will be the resulting distance? You mean that we pick $m_1, m_2, m_3, m_4$ such that

$$B=1+m_1+m_2+m_3 \\ R=2+m_4$$

where $m_1, m_2, m_3, m_4 \in \{ -1,1 \}$? #### Klaas van Aarsen

##### MHB Seeker
Staff member
You mean that we pick $m_1, m_2, m_3, m_4$ such that

$$B=1+m_1+m_2+m_3 \\ R=1+m_4$$

? Not quite.
Let's assume that after $i$ steps we have $B_i-R_i\equiv 1 \pmod 2$.
This is true for the initial position isn't it?

And let $B_{i+1}\equiv B_i +m_1+m_2+m_3 \pmod{12}$ and $R_{i+1}\equiv R_i+m_4\pmod{12}$, where $m_j =\pm 1 (j=1,2,3,4)$.
Then what are the possible values for $B_{i+1}-R_{i+1} \pmod 2$? Last edited:

#### evinda

##### Well-known member
MHB Site Helper
Not quite.
Let's assume that after $i$ steps we have $B_i-R_i\equiv 1 \pmod 2$.
This is true for the initial position isn't it?

And let $B_{i+1}\equiv B_i +m_1+m_2+m_3 \pmod{12}$ and $R_{i+1}\equiv R_i+m_4\pmod{12}$, where $m_j =\pm 1 (j=1,2,3,4)$.
Then what are the possible values for $B_{i+1}-R_{i+1}\equiv 1 \pmod 2$? Don't we want to have $B_{i+1}-R_{i+1}\equiv 0 \pmod 2$? #### Klaas van Aarsen

##### MHB Seeker
Staff member
Don't we want to have $B_{i+1}-R_{i+1}\equiv 0 \pmod 2$? Ah yes, I meant what are the possible values for $B_{i+1}-R_{i+1} \pmod 2$?
We want it to be 0 for a possible solution, but if we can prove that it's always 1, then that proves that there is no solution, doesn't it? #### evinda

##### Well-known member
MHB Site Helper
Ah yes, I meant what are the possible values for $B_{i+1}-R_{i+1} \pmod 2$?
We want it to be 0 for a possible solution, but if we can prove that it's always 1, then that proves that there is no solution, doesn't it? Yes, this would prove this. For the initial position we have $R_i-B_i \equiv 1 \pmod{2}$.

Suppose that $R_i-B_i \equiv 1 \pmod{2}$ for some arbitrary $i$.

Then $R_{i+1}-B_{i+1}=R_i+m_4-B_i-m_1-m_2-m_3 \equiv 1+m_4-m_1-m_2-m_3 \pmod{2}$.

How can we deduce that the latter is always equal to $1$ ? #### Klaas van Aarsen

##### MHB Seeker
Staff member
Yes, this would prove this. For the initial position we have $R_i-B_i \equiv 1 \pmod{2}$.

Suppose that $R_i-B_i \equiv 1 \pmod{2}$ for some arbitrary $i$.

Then $R_{i+1}-B_{i+1}=R_i+m_4-B_i-m_1-m_2-m_3 \equiv 1+m_4-m_1-m_2-m_3 \pmod{2}$.

How can we deduce that the latter is always equal to $1$ ? Consider all cases?
The shift in blue is one of $-3,-1,1,3$, and the shift in red is $\pm 1$ isn't it? Or not?
Then what are the possible changes in the distance between blue and red? #### evinda

##### Well-known member
MHB Site Helper
Consider all cases?
The shift in blue is one of $-3,-1,1,3$, and the shift in red is $\pm 1$ isn't it? Or not?
Then what are the possible changes in the distance between blue and red? It always holds that $R_{i+1}-B_{i+1} \equiv 1 \pmod{2}$.

We get for example the values $1+1-3=-1 \equiv 1 \pmod{2}, 1-1-3=-1 \equiv 1 \pmod{2}, 1+1+1=3 \equiv 1 \pmod{2}, 1-1-1 \equiv 1\pmod{2}$

and so on. Right? #### Klaas van Aarsen

##### MHB Seeker
Staff member
It always holds that $R_{i+1}-B_{i+1} \equiv 1 \pmod{2}$.

We get for example the values $1+1-3=-1 \equiv 1 \pmod{2}, 1-1-3=-1 \equiv 1 \pmod{2}, 1+1+1=3 \equiv 1 \pmod{2}, 1-1-1 \equiv 1\pmod{2}$

and so on. Right? Indeed. $\pm 3 \pm 1 = \pm 4, \pm 2, 0$ and $\pm 1 \pm 1 = \pm 2, 0$.
So the resulting change in distance is always even isn't it? #### evinda

##### Well-known member
MHB Site Helper
Indeed. $\pm 3 \pm 1 = \pm 4, \pm 2, 0$ and $\pm 1 \pm 1 = \pm 2, 0$.
So the resulting change in distance is always even isn't it? Yes, supposing that $R_i-B_i \equiv 1 \pmod{2}$, we get that $R_{i+1}-B_{i+1} \equiv 1 \pmod{2}$.

So, using induction, we have proven that $R_i-B_i \equiv 1 \pmod{2}$ for any $i$.

Thus, we deduce that the two balls will never meet at the same position.

Right? #### Klaas van Aarsen

##### MHB Seeker
Staff member
Yes, supposing that $R_i-B_i \equiv 1 \pmod{2}$, we get that $R_{i+1}-B_{i+1} \equiv 1 \pmod{2}$.

So, using induction, we have proven that $R_i-B_i \equiv 1 \pmod{2}$ for any $i$.

Thus, we deduce that the two balls will never meet at the same position.

Right?
Right! #### evinda

##### Well-known member
MHB Site Helper
Right! Nice, thanks a lot!!! 