Puzzles with numbers and things

[Obligation]

Some very good mathematicians present here!

Is the next question coming?

[davis_greatest]

Some very good mathematicians present here!

Is the next question coming?

Once I am home, tonight.

[Obligation]

Okie Dokie


Are you making these up yourself?

[davis_greatest]

Okie Dokie


Are you making these up yourself?

It is my pet orang-utan who makes them up, but he lets me post them and take his credit. He wasn't too pleased with me for saying he would ask a question about socks, but now assures me that he managed to think up a difficult sock question on his way home from work tonight. However, he's a bit hungry now and wants to go and have dinner first, before telling it to me for me to post a little later this evening.

[Obligation]

wow! you have a talented orang-utan!



is it for sale?

[April madness]

It is my pet orang-utan who makes them up, but he lets me post them and take his credit. He wasn't too pleased with me for saying he would ask a question about socks, but now assures me that he managed to think up a difficult sock question on his way home from work tonight. However, he's a bit hungry now and wants to go and have dinner first, before telling it to me for me to post a little later this evening.

I am getting suspicious of who of you to is doing the ironing!!!

[davis_greatest]

wow! you have a talented orang-utan!



is it for sale?

You'll have to discuss that with him directly.

[davis_greatest]

I am getting suspicious of who of you to is doing the ironing!!!

You think my orang-utan has time for ironing? What do you think he has me for?

[Obligation]

lol

[April madness]

You think my orang-utan has time for ironing? What do you think he has me for?

Cooking some dinner. I am sure he knows the fun of ironing

[davis_greatest]

Round 16 - Sock him one!

One year, after watching the about-to-regain-the-spot-of-world-number-one lose on a respotted black in the first round at the Crucible to Bond-Nigel-Bond, my orang-utan starts wondering if things could have been different. He thinks: what if a first round loser had potted just one more ball, would he have gone on to become world champion? What if? What if? Is the world champion really better than the first round losers, he wonders.

So, he organises a competition to find out. This is how it works:

He invites all the Crucible first round losers and the world champion to play in a league, round robin style, every player playing every other one once. At the end, after all the round robin matches have been played, the top two placed players will compete in a Final to determine the winner.

There will be one match each day, with a list of matches published in advance. In order to keep the distinguished look of the game, both players will have to wear socks the same colour as each other (i.e. only one colour - no odd socks allowed) during each match. The socks might be green, pink or ginger, decided randomly at the start of the tournament, and the match list will show, for each match, the colour of socks that will be worn that day.

My orang-utan and I decide, before the competition starts, that we definitely want to go to the Final. However, we don't want to look like muppets, so we decide that we will only go if we can go wearing the same colour socks as each other (i.e. only one colour - no odd socks allowed).

However, we find a few problems. First, we discover that we only have left socks, all of which we have already worn. However, my orang-utan soon solves this problem when he discovers that we can wear any sock on either foot.

Second, more worryingly, we discover that we only have 3 socks between us. I have 2 feet, and so does my orang-utan.

Question: prove that we don't have enough socks to go to the Final.

Only joking, that is not the question really.

Third, even more worryingly, we discover that our 3 socks are all different colours. One is green, one pink and one ginger!

To go to the Final will cost 70p for me. My orang-utan, as the real organiser and brains behind the tournament, can get in free to all matches. However, he will only go to a match if he can bring me along, since I can't be left at home alone. And I can't go alone because I won't know where to go.

To buy a new sock costs 20p, and we only have £1 in total, 70p of which we're going to need to go to the Final. Socks that have already been worn cannot be sold or swapped or sniffed (sorry, semih).

Each round robin match will cost 10p to see (for me; my orang-utan gets in free, remember). There is another slight problem - my orang-utan insists that he will only go to round robin matches provided he is not going to see more different players, in total during the round robin matches that he watches, than the number of round robin matches that he watches. (For instance, if he goes to 4 round robin matches, he doesn't want to see more than 4 different players in those matches.)

So, we wonder, how are we going to get enough socks to go to the Final?

My orang-utan has a think, and then has a stroke of genius. He has a lot of those. He declares, as organiser, that the winner of each match must remove his left sock at the end of the match and throw it into the crowd. He knows that no one else will grab it and, if we are there, we will be able to get it.

Show that, by studying the match list before the tournament starts, my clever orang-utan will be able to ensure that we will be able to go to the Final, beautifully socked.


PS No promises, but my orang-utan might even award 2 points for an answer to this question!

[snookersfun]

Sorry, can’t help myself. Especially one of these types of questions:

Not sure, if I understood the whole problem correctly though. Let me put it in numbers:

16 first round losers + 1 champion = 17 players
therefore 136 round robin games
45, 45, 46 of those played in each color of socks, 1 each up for grabs after the game

you and monkey together have only 1 of each sock + 1pound (-70 p for the final), need three more


suggestion: go to three round-robbins (30 p) of the same color to find three more same colored socks

problem: out of these 46 matches, you’ll have to find three combinations, such that only three players are involved, such as 1,2-1,3-2,3

seems to me that 16+15+14 is the max. # of games, which can involve only pairings. Thus, the moment, you choose the group with the 46 colors, you are bound to find your pairing.

[davis_greatest]

Sorry, can’t help myself. Especially one of these types of questions:

Not sure, if I understood the whole problem correctly though. Let me put it in numbers:

16 first round losers + 1 champion = 17 players
therefore 136 round robin games
45, 45, 46 of those played in each color of socks, 1 each up for grabs after the game

you and monkey together have only 1 of each sock + 1pound (-70 p for the final), need three more


suggestion: go to three round-robbins (30 p) of the same color to find three more same colored socks

problem: out of these 46 matches, you’ll have to find three combinations, such that only three players are involved, such as 1,2-1,3-2,3

seems to me that 16+15+14 is the max. # of games, which can involve only pairings. Thus, the moment, you choose the group with the 46 colors, you are bound to find your pairing.

I will have a look at this a bit later when I can grab more than 1 minute to see if I am convinced! Can you explain the last bit... "seems to me that 16+15+14 is the max. # of games, which can involve only pairings"?

And he's an orang-utan, not a monkey!

[snookersfun]

I will have a look at this a bit later when I can grab more than 1 minute to see if I am convinced! Can you explain the last bit... "seems to me that 16+15+14 is the max. # of games, which can involve only pairings"?

And he's an orang-utan, not a monkey!

Sorry for having offended your friend

OK, then, I tried to go opposite on that last point, i.e. what are the maximum possible pairings, so that no three pairs made up of three players could be in the same socks.
So one could put all the pairs including player A in green socks (16 pairings), after that all the players (naturally excluding A) including B in pink (15 pairings) and the third biggest group with player C in ginger (14 pairings). So far (45 games) you have avoided giving three such pairs the same color of socks.
Now, the moment any other player plays anybody else, it has to be one of the three colors again and accordingly the third pair is found.

[davis_greatest]

Sorry for having offended your friend

OK, then, I tried to go opposite on that last point, i.e. what are the maximum possible pairings, so that no three pairs made up of three players could be in the same socks.
So one could put all the pairs including player A in green socks (16 pairings), after that all the players (naturally excluding A) including B in pink (15 pairings) and the third biggest group with player C in ginger (14 pairings). So far (45 games) you have avoided giving three such pairs the same color of socks.
Now, the moment any other player plays anybody else, it has to be one of the three colors again and accordingly the third pair is found.

My orang-utan has been scratching his head, something he likes to do a lot, but is unfortunately still not convinced.

I think the first part of your answer showed that there must be A set of at least 46 round robin matches where the socks thrown into the audience are all of the same colour. Is that right?

In the 2nd part, are you arguing here that if we take that set of at least 46 matches (which could, of course, be ANY combination of different matches), my orang utan must be able to find among them a set of 3 matches such that only 3 players are involved?