Physics behind "The Wall" Game show

[Chand3994]

In "The Wall" game show, the slots and also the diverters are designed symmetrically and also identically.  So when a ball is dropped from a particular slot number it should end up in a particular amount.  But how does the ball end up falling into different amounts on other subsequent tries even when the ball is dropped from the exact same position. The entire wall is glass enclosed so there is no air resistance.   The ball is also dropped by machine,no manual input hence equal force of drop in every try. I don't know which factor affects the ball as it changes its path in every iteration even when it is dropped from the same slot. Can someone explain the science behind this on why the ball takes different route each time? 

[Strange]

Good question.

This is an example of "chaos theory" (popularly known as the butterfly effect) in action. This tells us that very tiny changes in initial conditions can lead to very different outcomes, even in completely deterministic systems like the wall.

If every ball were perfectly spherical and perfectly smooth, and if every one were exactly the same mass and dropped at exactly the same position and exactly the same speed, and there were absolutely no outside influences (vibration, changes in temperature, noise, etc.) then they should follow the path every time.

But none of those things are true. The balls are not perfect, they are not completely identical, they will drop from very slightly different positions at very slightly different speeds, etc. So they will bounce off the first pin at very slightly different angles. Then with each subsequent bounce the difference is magnified (because any tiny change in the angle the ball hits a pin will cause a larger deviation in the direction it bounces off).

More here: https://www.forbes.com/sites/startswithabang/2018/02/13/chaos-theory-the-butterfly-effect-and-the-computer-glitch-that-started-it-all/#7f140cc069f6

4 hours ago, Chand3994 said:

The entire wall is glass enclosed so there is no air resistance.

Unless there was a vacuum behind the glass, there will still be air resistance. But there won't be effects from external air movement (competitors blowing at it!)

Incidentally, the original design of this thing is well over 100 years old and was designed to demonstrate a different mathematical principle: https://en.wikipedia.org/wiki/Bean_machine

[Ghideon]

4 hours ago, Chand3994 said:

 But how does the ball end up falling into different amounts on other subsequent tries even when the ball is dropped from the exact same position.

I would say that due to miniature imperfections in the balls, the dropping mechanism and the pins. If we look at the first pin, there will be small fluctuations in force and angle of the drop so that it is 50/50 percent change for the ball to go left or right. If the imperfections are small enough the left or right path is not predictable for a single ball. Any small imperfections add up along the balls' path making each pin more or less a 50/50 chance. 
If there would be too much imperfection the odds would be off, for instance if the machine always would drop the ball too far to the left or right. One should try to build the drop with good* precision.
This explanation assumes that the machine is built to be random. If not, there are numerous options. The pins could be spinning or vibrating, There could be an airflow behind the glass front. And lots of others.

So far we have discussed one ball. Over a large number of balls the average outcome is very predicable. Unless machine is constructed to be non-random a large numbers of balls will approximate a normal distribution**. This means that there will be more balls in the centre slots than at the edges.

After dropping hundreds of balls in the wall machine the piles would look similar to this:

Does this answer your question? From here we could discuss the math and its implications on the game setup. Or mechanical aspects of the machine. There are also 

 

Some notes:

4 hours ago, Chand3994 said:

The entire wall is glass enclosed so there is no air resistance

That requires that the machine is built as vacuum chamber. 

 

4 hours ago, Chand3994 said:

The ball is also dropped by machine,no manual input hence equal force of drop in every try.

We can assume the machine is built that way. There could be a random generator connected to the drop mechanism allowing it to change the force by some small amount, but not enough to be visible.

 

*) If the precision were 100% perfect all the balls could possibly land exactly head on the first pin, bounce up and down and finally come at rest there. The machine needs (and will always have) some tiny imperfections.

**) Example machine and the math is discussed in https://en.m.wikipedia.org/wiki/Bean_machine 

 

 

edit: x-post with @Strange. And no, we did not plan to create identical answers  

 

Edited January 26, 2020 by Ghideon
x-post explained

[Strange]

Great minds, eh.

[swansont]

2 hours ago, Ghideon said:

 

So far we have discussed one ball. Over a large number of balls the average outcome is very predicable. Unless machine is constructed to be non-random a large numbers of balls will approximate a normal distribution**. This means that there will be more balls in the centre slots than at the edges.

After dropping hundreds of balls in the wall machine the piles would look similar to this:

 

one can buy these - Google “Galton board”

I have one in my office. 

[Chand3994]

You guys have explained it really well with physics and statistical background. Really amazing. Thank you very much @Strange @Ghideon @swansont

[Phi for All]

6 hours ago, Ghideon said:

If the precision were 100% perfect all the balls could possibly land exactly head on the first pin, bounce up and down and finally come at rest there.

If you could fake a video of exactly this, it would be a good perspective on the difficulty in removing chaos from a system. With the proper buildup, it would also be hilarious. 

[Strange]

Also, I wonder if the balls are dropped from the same point every time. There are those penny in the slot machines where you drop coins in, hoping to displace an accumulating pile of previous coins. The mechanism swings backwards and forwards so coins never go quite where expected. The Wall might do a similar thing, just to add to the randomness.

Or maybe it isn't a machine at all but some guy up a ladder, throwing balls through the holes!

[Phi for All]

How difficult would it be to design an electromagnetic system behind the wall so the ball ends up in the same place no matter where it starts? With current technology, could it be made to look natural?

[Strange]

Just now, Phi for All said:

How difficult would it be to design an electromagnetic system behind the wall so the ball ends up in the same place no matter where it starts? With current technology, could it be made to look natural?

Interesting question. I just watched a video of the ball falling. It looks like it is very light (foam?) and bounces quite a lot. So little jets of air at each pin might be a better method. The bouncing could hide an extra unexpected movement.

The interesting thing is that you would not need to successfully deviate the path at every pin - the angle it falls in some places would make that too difficult. Instead a computer could track its path and redirect it in the right general direction at the next pin.

I don't know if you could always get it in the same place, but you could skew the odds massively.

Another approach that might work is spinning the pins to give some extra impetus one way or the other.

[Phi for All]

3 minutes ago, Strange said:

It looks like it is very light (foam?) and bounces quite a lot. So little jets of air at each pin might be a better method. The bouncing could hide an extra unexpected movement.

Very light foam around a small iron ball?

4 minutes ago, Strange said:

The interesting thing is that you would not need to successfully deviate the path at every pin - the angle it falls in some places would make that too difficult. Instead a computer could track its path and redirect it in the right general direction at the next pin.

That's what I was thinking. Not sure how sophisticated it would need to be, but I imagine the technology is available now.

8 minutes ago, Strange said:

Another approach that might work is spinning the pins to give some extra impetus one way or the other.

 A computer might be able to use all these forces together to achieve a fairly natural-looking illusion of randomness while actually controlling the ball to a fair degree.

[Ghideon]

1 hour ago, Phi for All said:

A computer might be able to use all these forces together to achieve a fairly natural-looking illusion of randomness while actually controlling the ball to a fair degree.

I have zero experience from such construction but it is a fun question!
Intuitively I think of a light ball. But not too light, it needs some momentum so that adjustments to its path does not look too obvious. Then I would drill holes along the length of the pins the ball bounces off, one row of holes in each pin. The pins will be individual rotatable and connected to individual air flow controls. That means that any pin can blow air in any direction, individually. Finally I would try to program the pins airflow and direction so that the active pins form a "funnel". At the top, where the ball is dropped there is no airflow. Then along the desired path from top to bottom more and more pins are blowing air slightly towards the centre of the desired path. That makes the first few bounces random and then the control kicks in. If the airflow is gradually stronger further away from the paths that lead to the desired slot the ball will have a very limited chance of bouncing there. 

Illustration, X is desired slot and arrows is active airflow. 

 

Edited January 26, 2020 by Ghideon
picture added

[Chand3994]

8 hours ago, Strange said:

Another approach that might work is spinning the pins to give some extra impetus one way or the other.

The Game show already has eyes of great minds all over the world looking for evidence to prove that they are controlling the balls manually as per the show owner's wish thereby telling that the show is fake.   So any sort of movement of the pins or holes in pins would make a strong evidence as everything is visible in that wall.  That's the reason maybe why all pins are stationary in a observers perspective so as to not give any clues to the observer to prove that this show is fake.  So any kind of alteration in the pins would defeat this purpose

[Ghideon]

5 hours ago, Chand3994 said:

The Game show already has eyes of great minds all over the world looking for evidence to prove that they are controlling the balls manually as per the show owner's wish thereby telling that the show is fake.   

I don't think we intended to claim that the show was fake, just discussing if it was possible to engineer such a machine. But your concern raises an interesting related point of view. As said above each ball is random, allowing for the show creators to utilize these moments of "luck" or "excitement" in the show. But since the outcome, in the long run, is mathematically proven I think the network that run the show can make solid calculations about the payout of prize money. I haven't tried the math but given the number of balls per show, the number of slots, the payout of each slot, the number of shows per season (and maybe some more) one could predict the budget for the prizes for one season of show. So given that there are enough shows planed there is not so much need for fakes, at least not to "protect" the shows budget.
(There could of course be other reasons to try to affect the odds.)

[swansont]

If the balls were ferromagnetic, one could alter the path with magnets behind the board. Strong ones would not work, as the deviations would be obvious. But you could skew the probabilities. Or subtle shaping of the pins - the deflection not being 50/50

 

But it's probably not necessary. This is a game of chance, and in the long run, we know the outcome. All the show has to do is make sure its revenue exceeds its cost, like any game show. No different than running a roulette wheel at a casino. The house has a several percent advantage, and even though it pays out people getting the number right, it takes in more than it pays out in the long term.

[Strange]

In terms of managing the probabilities of payouts, the show can also adjust the number and amounts of the prizes in the "bins" at the bottoms of the wall. 

[Chand3994]

On 1/27/2020 at 3:20 PM, Ghideon said:

I think the network that run the show can make solid calculations about the payout of prize money. I haven't tried the math but given the number of balls per show, the number of slots, the payout of each slot, the number of shows per season (and maybe some more) one could predict the budget for the prizes for one season of show

@Ghideon The payout can be approximated by what is called Pascal's Triangle and Binomial Distribution.  More details on Galton Board video here

On 1/27/2020 at 4:58 PM, Strange said:

the show can also adjust the number and amounts of the prizes in the "bins" at the bottoms of the wall. 

They did actually.  Pascal's Triangle States that a ball dropped from a particular slot has higher probability of falling into the bin right below it.  This is the reason why the maximum money bin which is at the far right does not have any slot directly above it.  The Wall structure is curved at the top in order to reduce the probability of getting to maximum money bin value :D

Edited January 29, 2020 by Chand3994
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