Puzzles with numbers and things

[davis_greatest]

Question 12 – Pot Smack Marathon – The Clear-Up

The Pot Smack Marathon is over, but Petdon is not happy. He potted most balls and Grott the fewest, yet Petdon did not get a chance to give Grott a single smack – something he had been itching to do since May.

Shirty is also not happy. He also potted more balls than Grott, yet his moonface received over 148 million more smacks than Grott received.

Petdon and Shirty’s father, Tony, who both happen to be on the board of the WBPSA (World Billiards and Pot Smack Association) write letters of complaint to the WBPSA disciplinary committee about the two Pot Smack Marathon organisers, my friend Clive and me.

The Board decide to resolve it in the following way, which will give Petdon the chance to give Grott many smacks. First, each of the 3 competitors is given some of the balls that he had potted during the Pot Smack Marathon, each player being given balls of one colour.

Grott, who potted quite a few yellows over the autumn, is given 1,000 yellow balls. Shirty is given 1,000 black balls and Petdon, who potted the brown many times over the autumn, is given 4,000 brown balls.

This is what will happen. The following day, each player will be given a score depending on the total value of his balls (e.g. Grott’s score is 2,000 as he has 1,000 yellows). Petdon will be permitted to give Grott one smack for every point by which Petdon’s score exceeds Grott’s. E.g. If Petdon has a score of 20,000, he will be able to give Grott 18,000 lovely smacks.

Petdon and Shirty strike up a little deal. Petdon just wants to maximise the expected number of smacks he will give Grott. Shirty wants to end up with as many balls as possible, but doesn’t care about their colours or value as he is not going to give or receive any further smacks.

This is how the deal will work: A random number from 0 to 1,000 inclusive will be drawn by lots by Clive (each number from 0, 1, 2, …, 1000 being equally likely). Whatever number is drawn, Shirty will put this number of his black balls in a trunk. Shirty and Clive know the number, but Petdon does not.

Petdon is then required to put as many of his own brown balls as he chooses into his own trunk. This will be passed to Shirty for inspection. If Petdon has offered more brown balls than (or as many as) there are black balls in Shirty’s trunk, Shirty will accept them and exchange trunks. Otherwise, there is no deal – Shirty will keep his black balls and Petdon will get his brown balls back.

Petdon asks you for advice on how many brown balls to offer in the trunk. You want to help Petdon in his aim of maximising the expected number of smacks he gives Grott. How many brown balls should you tell Petdon to offer?

[snookersfun]

is that statistics? I hate statistics
it's below 1000 isn't it?

[davis_greatest]

is that statistics? I hate statistics
it's below 1000 isn't it?

Yes, this is related to mathematical statistics, but you don't need any knowledge of the subject to solve it. Just a bit of logical thinking

It MIGHT be below 1,000. Not saying yet.

[snookersfun]

I don't know, wouldn't offer any balls. Seems to me, the probability (or rather the amount of points) to loose is always bigger than the possible gain.
But that's probably not so, is it...

It MIGHT be below 1,000. Not saying yet.

yeah, right

[davis_greatest]

Well, I think you're pretty much there, snookersfun!

Now I just need an explanation to demonstrate that you are right

[snookersfun]

ohmegod, didn't really expect that... you're not kidding, are you?

Ok, I try to throw something

there are the two options
a) he puts too few balls, no exchange, nothing lost
b) he puts enough balls say x+a (x being the initial # drawn and a the #s added over that value), thus your possible win would be 3x (the difference in value of the balls) upset by any 'overpaying' -4a (as he would just loose any of his overpaid brown balls).
And now you better explain the rest of it...

[davis_greatest]

ohmegod, didn't really expect that... you're not kidding, are you?

Ok, I try to throw something

there are the two options
a) he puts too few balls, no exchange, nothing lost
b) he puts enough balls say x+a (x being the initial # drawn and a the #s added over that value), thus your possible win would be 3x (the difference in value of the balls) upset by any 'overpaying' -4a (as he would just loose any of his overpaid brown balls).
And now you better explain the rest of it...

Interesting - however, I will give some more time either for you to prove what you have said or to give someone else time to have a go I don't think you've yet shown that the overpaying is more than the win...

E.g. If Shirty has 400 black balls in the trunk, and Petdon offers 500 brown balls, then Petdon gets 2800 points from Shirty at a cost of 2000 points, so in that case it was worth doing.

[snookersfun]

...so now in order to gain, 'a' has to be smaller than 3/4 x, thus there are simply more possible numbers leading to a loss than there are numbers resulting in a potential gain...

...well more like, allthough there is an equal possibility of winning and loosing, combined or max. possible point loss is worth 4/3 of a possible gain...

give someone else the chance, have to do Algebra homework with the kids now

[davis_greatest]

Hall of Fame and Scoreboard (after round 11)

Well, maybe I could have started this earlier, but better late than early. Here is the Puzzles with numbers and things Hall of Fame and Scoreboard.


Question 1 – Snooker League. Congratulations robert602!

Question 2 – Cue in the Attic. Congratulations robert602!

Question 3 – Bad Witches. Congratulations robert602!

Question 4 – Let’s play Nookers! Congratulations robert602!

GREAT START FROM ROBERT602– HE’S LOOKING UNSTOPPABLE! 4-0

Question 5 - 962,537,588,106-Ball pool. Congratulations abextra!

Question 6 - Joining the dots... or squaring the circle?. Congratulations abextra!

WHAT’S THIS? ROBERT602 HAS FALTERED AND ABEXTRA IS STORMING AFTER HIM. ROBERT602 LEADS ABEXTRA 4-2

Question 7 - Snooker Tricolore. Congratulations Vidas!

NEW KID ON THE BLOCK VIDAS SHOWS THEM HOW IT’S DONE WITH THE TOILET !

Question 8. – Counting on fingers. Congratulations snookersfun!

Question 9 – Pot Smack. Congratulations snookersfun!

SNOOKERSFUN COMES ON THE SCENE AND STARTS A CHALLENGE

Question 10 – Pot Smack - The Morning After. Congratulations part a) abextra and b) Vidas again!

Question 11 – Pot Smack Marathon. Congratulations snookersfun!

SNOOKERSFUN CONTINUES THE CHARGE!

Question 12 - Pot Smack Marathon – The Clear-Up. Still open..... although snookersfun MAY have done enough for half a point! However, this has not been awarded yet in the scoreboard below, as it’s not guaranteed!


-------------------------------------------------------------------


SO HERE IS THE SCOREBOARD AFTER ROUND 11

robert602…………………………………4
snookersfun……………………….…..3
abextra……………………………..…...2½
Vidas……………………………………….1½

[snookersfun]

oh, so you do dole out points!

So for another 1/4 point then.
looking at the two possible extremes, shows the same math as above:
a) 1000 black balls in and 1000 brown balls offered - a win of 3000 points
b) 0 black balls in and 1000 browns offered - a loss of 4000 points

or another way:
e.g. 200 brown balls offered, but
a) no black in - loss of 800 points
b) 200 blacks in - win of only 600 points

this could be continued for all possible scenarios (x and a), thus generally winnings only 3/4 of possible losses.

[davis_greatest]

Another point for snookersfun!

OK! I wouldn't call that a formal proof, but I think you can see enough of what is going on that I would just feel mean withholding the point!

Here is an explanation:

Suppose Petdon offers x brown balls, with a points value of 4x.

If Shirty has more than x black balls in his trunk, there will be no deal, and Petdon neither loses nor gains from the deal.

Given that Shirty has no more than x black balls in his trunk and that the deal is done, there must be an equal probability of each number of black balls from 0, 1, 2, ..., x being in Shirty's trunk. Therefore, given that Shirty has no more than x black balls in his trunk, the expected number of black balls is x/2. As each ball is worth 7, their expected value is 7x/2. Petdon gives balls of value 4x for these.

In this case, given that the deal is done, Petdon's expected profit would be
7x/2 - 4x = -x/2. Hence, Petdon's expected profit is negative, unless x=0, i.e. he offers no balls.

So your advice to Petdon is not to bother.


SO HERE IS THE SCOREBOARD AFTER ROUND 12

This scoreboard works slightly differently from those in the other quizzes. Where points are tied, the person placed top in the tied group is the one who scored his or her last point FIRST. I think that that is arguably slightly fairer. So, snookersfun, I'm afraid you're still in second place!


robert602…………………………………4
snookersfun……………………….…..4
abextra……………………………..…...2½
Vidas……………………………………….1½

[snookersfun]

phew! the whole point also, very generous...

as always your proof is quite more elegant and complete (have to tell you I am rather rusty )

I don't mind being 2nd (unless Curtis will find us a sponsor), until a few minutes ago, there were no points, so I am happy.

[davis_greatest]

Round 13 - Pot Smack - The Revenge

After the Pot Smack clear-up, Clive and I still aren't best pleased that Ebony Petdon and Maun Shirty's father Tony wrote letters of complaint about us to the WBPSA, but to show there are no hard feelings, we have a couple of friendly games of snooker that evening.

First, I play Shirty. Shirty knocks in a 40 break, then misses a red. I then knock in a 60 break, finishing with the final pink to win the frame.

"Listen, Maun," I say, "seeing as I just scored 1½ times your score, if I can find a (positive whole) number such that when I take the last digit off and put it on the start, it is 1½ times the original number, can I give you that many smacks?"

"OK," he says. "As long as it is the smallest such number possible."

"Fine," I say. "285,714 then. Look, if I take the 4 off the end and put it on the start, I get 428,571, which is exactly 1½ times the original number."

Well, Shirty looks long and hard, but has to concede that I have found the smallest such number, so I give him 285,714 smacks.

Then, Clive plays Petdon. Petdon knocks in a 50 break, then misses a red. Clive knock in a 75 clearance to win the frame.

"Petdon," says Clive, calling him by his surname, "seeing as I just scored 1½ times your score, if I can find a (positive whole) number such that when I take the first digit off and put it on the end, it is 1½ times the original number, can I give you that many smacks?"

"OK," Petdon says. "As long as it is the smallest such number possible."

Clive, being pretty sharp, does a few calculations, and not long later he is starting to give Petdon very, very many smacks.

How many smacks will Petdon receive?

[snookersfun]

I am suffering....

can give you the last 10 digits so far
if you tell me, that I havn't much longer to go, I ,ight consider going on

[davis_greatest]

Last 10? Let's see them then!