Super-impossible math questions.

Mr know-it-all to the right has has carefully planned for us this section dedicated to bringing you the world's hardest math/logic questions. The questions are such that even a ten year old could understand them, but it would take more than a nuclear scientist to solve them! The idea is not to solve them, but to be confident in the knowledge that no-one else could solve them (unless given masses of time and a super computer - and probably not even then either) - thus being blissfully ignorant that anyone has even bothered to try and work them out at all. Also, this gives Mr Know-it-all a huge ego boost as Mr Know-it-all and /only/ Mr Know-it-all has the answers.
Actually, all this is a lie; the real intention behind this section is to humiliate you, and to humiliate you big-time. Seriously, for the few who do attempt to work them out, I wish you luck, and please, please send in your answers to my email so that I can show Mr-know-it-all and possibly deflate his ego just a little.


OK, OK, I'll start with one or two 'easy' ones just to 'prepare' you for the killers later.

The dice roll:
"oooh... that's just impossible" rating: 1.4/10

If you throw a dice 6 times, what's the chance that you'd get a six on:
a: exactly one of the die.
b: one or more of the die.




The infinity thing:
"oooh... that's just impossible" rating: 4/10

Starting at number zero, you throw a coin.
Tails means you go down one (therefore equalling -1)
Heads means you go up one (therefore equalling 1)

You repeat this precedure infinite times, so here's just one possible variation:
0 1 2 3 2 3 4 3 2 3 4 3 4 5 6 7 6 7.........

Question is: - how many times on average would 0 have been 'hit' if you threw the coin 100 times? This question is actually possible to answer, but you might need to use a computer ;) The highest possible answer is 50 (0, 1, 0, 1, 0, 1 etc.), and the lowest answer is 0 (many, many ways of getting this), but I want the /average/.

Now the killer /when even a computer/ won't be much help (impossibility sub-rating 6/10)

What's the average amount of times you'd have to throw the coin before zero was hit upon again?
I know this sounds simple, but there's a cunning nature to this question. For example, the coin tosses could go all the way down to -5000 and then suddenly 'decide' it wants to edge its way back up (albeit it in a stuttering way) to 0.


---------------------- The zigzagger:
"oooh... that's just impossible" rating: 5/10

10 metres away from someone is a circle with a diameter of 1 metre. The idea is that he chooses a completely random number from -90 to +90 (relative to the target) and uses this number as his direction in degrees.
Each 'occasion', he walks 2 metres, before he has to stop and take a break. Of course, he may easily miss his target (and since there is no going back thanks to only 180 degrees of movement), what's the chance he will hit it?
See left for a brown diagram showing the various lucky possibilites that hit the target.

That to easy for you? ok....

Now the killer (as if that previous one wasn't hard enough) (impossibility sub-rating 6.5/10):

The random direction now extends to 360 degrees. Thus he could therotically take 1000 years (or 1,000,000 'occasions') to hit it, but it is possible. What's the chance of hitting the green target on the thousandth attempt?
See right for a purple diagram showing one of the various lucky possibilites that hit the target.

Next horribly, horribly complicated question


The cricket pitch:
"oooh... that's just impossible" rating: 6/10

On a cricket pitch measuring 40*25 metres, a bowler bowls to a batsman who hits a ball at a certain speed. There are 20 'catchers' (placed at the points indicated by the diagram to the left), who each cover a small section of the pitch: What is the likelihood of the ball being caught if all of the following criteria are met:

a: the moment the ball has been hit, the catchers 'magically' know where the ball will end up, and start moving to that position at a constant speed.
b: the catchers always catch the ball if they can reach it.
c: the batter hits the ball such so that it only lands on the patterned green area, and that each 'spot' of this pitch is just as likely to be reached as any other 'spot'.
d: The ball travels at a speed of 10 metres per second.
e: The catchers run at different speeds, with the ones at the back being the slowest. Here are the speeds each row of catchers possess (mps=metres per second):
Row a: 1 mps
Row b: 1.5 mps
Row c: 2 mps
Row d: 2.5 mps
Row e: 3 mps

And the killer question (impossibility rating 7/10):

Same as above, except if two or more players manage to reach the ball, they both get confused about who should catch the ball, and thus neither catch it. What is the likelihood of the ball being caught under these circumstances now?

The curvy rebound:
An easier version of an angle prob to prepare you for what's to come.

"oooh... that's just impossible" rating: 7/10

An infinitely small ball is placed at a random point on the red line shown on the right. The line is 10 metres long. The semi-circle that stems from this line is its resulting size too. Also positioned 3 metres above the centre of the line is a small circle with a diameter of 1 metre. There is no gravity of any sort by the way.

The ball is shot up into the semi-circle at a random angle from -90 degrees to +90 degrees. After it comes back past the red line, it is all over. What's the chance the 'ball' will contact the yellow circle?

Same question as above, except that the curve (which has now extended to become a full circle) rotates (axis = point at centre of yellow circle) 45 degrees for every 5 metres the ball has travelled (impossibility sub-rating 8.5/10)


----------------------------------------The starry rebound:
"oooh... that's just impossible" rating: 7.5/10
They get harder....

An (infinitely small) ball starting out in the middle of a 5 pointed star table (outer 5: 10m radius..... inner 5: 5m radius) has a starting angle of a random value from 0 to 360 degrees. The ball is now set loose and travels around the table.
On average, how many sides will have been hit once the ball has travelled 100m ?

And the killer question: (impossibility sub-rating 8/10)
The ball has now got a 'real' size of 1m diameter and thus is affected in weird ways by colliding and bouncing off at tangents with the 'inner' 5 points etc.

Now how many sides on average will have been hit once the ball has travelled 100m ?
Also, where are the most likely points that the ball will end up?






The nightmare of nightmare questions:
The snooker table of doom:
"oooh... that's just impossible" rating: 9.5/10

A 'snooker' table (measuring 8 metres by 4m) with 4 'pockets' (measuring 0.5m and placed at diagonal slants in all 4 corners) contains 10 balls (each with a diameter of 0.25m) placed at the following coords:
2m,7m...(white ball)
...and red balls...
1m,5m... 2m,5m... 3m,5m
1m,6m... 2m,6m... 3m,6m
1m,7m... 2m,7m... 3m,7m

The white ball is then shot at a random angle from 0 to 360 degrees.

Assuming the balls travel indefinitely (i.e. no loss of energy via friction, air resistance or collisions), answer the following:

a: What exact angle should you choose to ensure that all the balls are potted the quickest?
b: What is the minumum amount of contacts the balls can make before they are all knocked in?
c: Same as b, except that each ball that is knocked in must not have hit the white ball on it's previous contact (must be a red instead).
d: What proportion of angles will leave the white ball the last on the table to be potted?



All answers