Puzzles with numbers and things

[snookersfun]

lowest breaks hopefully 6587.
I will look over it again. I just only found the easy way (I am keeping waking up with those inspirations)... Have to rush to work... more later

[davis_greatest]

Note: I have just changed the last line of the question above to ask something slightly different - I had originally asked "What were the joint lowest breaks of the night?"

Anyone who had already read the original question and prefers to answer that instead will, of course, get the point for answering it. (If you can answer one then you can almost certainly answer the other.)

[davis_greatest]

Round 48 - Monster, Maxi, Big and Small

Before reading this, you need to read the explanation of Monstersnooker and Maxisnooker in Round 47 (message 588) http://www.thesnookerforum.com/showt...2-page_59.html

================================================== ========

"You know," said Gordon to Oliver the following day. "Charlie and I had some great games of Monstersnooker and Maxisnooker last night. You should have been there."

"It was very confusing, though," said Charlie. "We found out that there are actually two types of Maxisnooker: Little Maxisnooker and Big Maxisnooker."

"Really? What happened?" asked Oliver.

"Well," said Charlie. "First we found that Gordon had ordered 9 sets of reds from Little Maxisnooker. So, we put the 9 sets of reds together with the red that I had brought (making a perfect triangle!), and played a frame of Monstersnooker with them. I knocked in a fantastic total clearance, with pinks and blacks with all the reds."

"Yes," said Gordon. "And then we played 8 frames of Little Maxisnooker. And I won them all. I made 8 maximums!"

"So," said Charlie. "Gordon was beating me 8-1. But we weren't counting frames. We were counting total points. And, after those 9 frames (1 Monstersnooker and 8 of Little Maxisnooker), we were exactly level in aggregate points! Amazing!"

"Then," said Gordon, "the confusion began. Because then 9 sets of reds from Big Maxisnooker arrived. So we used those balls instead. We found, when we saw the balls, that there is exactly one more row of reds in the triangle in Big Maxisnooker than in Little Maxisnooker."

"Yes," said Charlie. "So, we put the 9 sets of reds from Big Maxisnooker together with the red that I had brought (again, of course, making a perfect triangle), and played a frame of Monstersnooker with them. Of course, this frame of Monstersnooker had more reds than the earlier frame of Monstersnooker. This time, Gordon knocked in a total clearance, with pinks and blacks with all the reds. So Gordon was leading 9-1 in frames."

"Yes," said Gordon. "And then we played 8 frames of Big Maxisnooker. And this time bloody Charlie won them all! And - guess what? HE made 8 maximums!"

"And what's more," said Charlie. "It meant that not only was the frame score 9-9, but we were still level on aggregate points! 18 frames played, 18 total clearances, including 16 maximums, and no fouls!"

"But I had the more fun last night," said Gordon.

"Why?" asked Oliver.

"Because," said Charlie with a wry smile. "Gordon potted the pink 200 times more than I did."


How many rows of reds in a triangle in Big Maxisnooker?

[snookersfun]

Well, yes, I should weed out these typos or read over my answers or whatever.
I liked the triangle above specifically, as I was pondering for a while, why, while one can geometrically form a bigger triangle exactly from 9 smaller equal ones, one has to add the one ball here (of course one would have to share the corner balls and probably could prove the result this way as well).

[davis_greatest]

HERE IS THE SCOREBOARD AFTER ROUND 47 after awarding abextra and snookersfun 1 point each:

snookersfun……………………….…..20
abextra……………………………..…...11
Vidas……………………………………….8½
davis_greatest…………………..……5½
robert602…………………………………5
elvaago...............................4
Semih_Sayginer.....................½

(some rounds may be worth more than one point)
(especially ones won by davis_greatest)

[davis_greatest]

Congratulations, abextra and snookersfun!

1 point to abextra for the correct answer of one red ball, and 1 point to snookersfun for subsequently sending me the proof by Private Message (with just a minor error that I think was a typo).

No matter what the size of the triangles in Maxisnooker, Charlie only needs to bring one red ball.



It’s not very easy to draw on here, but the diagram below is intended to show that if you take 9 equal triangles of balls, you can always add one more ball (shown as an orange X in the centre) and make a bigger triangle. I hope you like this, because this artwork below took me about 100 times longer to produce than to come up with the question!


......................O
....................O O
...................O O O
.................O O O O
...............O O O O O
.............O O O O O O
...........O O O O O O O
.........O O O O O O O O
.......O O O O X O O O O
.....O O O O O O O O O O
...O O O O O O O O O O O
.O O O O O O O O O O O O
O O O O O O O O O O O O O

This is an example with triangles of 4 rows, but the principle is the same for any size triangles.


Algebraically, we can show this as follows:

If a triangle of reds in Maxisnooker consists of m rows, then the triangle has m(m+1)/2 red balls.

Gordon brings 9 sets of Maxisnooker triangles, i.e. 9m(m+1)/2 red balls.

If you add one ball from Charlie, then the number of red balls is

9m(m+1)/2 + 1

= [9m(m+1) + 2] / 2

=[9m^2 + 9m + 2] /2

=[(3m+1)(3m+2)] /2

which is the number of reds in a triangle with 3m+1 rows!



Scoreboard to follow....

[abextra]

I'm afraid, I can't! I used formula N=n(n+1)/2 (N - total number of balls in a triangle, n - number of balls at one side) and some smaller numbers to find out my answer of one ball, but I have no idea, why do we have to add this one ball to nine sets to form a perfect triangle!
So, if anyone else can give a proof, have a go!

[davis_greatest]

Yes, it would! Can you prove that one ball will always be enough?

[abextra]

point AND A HALF, abextra!

Special thanks for the HALF point too!

Round 47 - would 1 ball be enough?

[davis_greatest]

For anyone who's already read round 47 in the couple of minutes since I posted it, please note that I have updated the wording of the last 2 or 3 paragraphs to make it a bit clearer - so please re-read the end.

[davis_greatest]

Thank you for the point, davis_greatest, I'm glad I didn't break the table.

point AND A HALF, abextra!

[davis_greatest]

Round 47 - Monstersnooker

"Want a game of Monstersnooker tonight?" Gordon asked Charlie.

"Monstersnooker?" asked Charlie. "What's that?"

"I invented it," replied Gordon. "It's just like normal snooker. But the triangle of reds is bigger - more than the 15 reds you get in snooker."

"Oh? How many reds in the triangle?" asked Charlie.

"I don't know!" said Gordon.

"You don't know, Gordon? Then where are you getting the reds from?"

"Well," said Gordon. "I've ordered 9 sets of red balls used for a normal game of Maxisnooker. I'm taking them along tonight."

"Maxisnooker?" asked Charlie. "What's that?"

"Oh," said Gordon. "It's just like normal snooker. But the triangle of reds is bigger - more than the 15 reds you get in snooker."

"Oh? How many reds in the triangle in Maxisnooker?" asked Charlie.

"No idea! I don't know!" said Gordon.

"You don't know?! You don't know, Gordon?! Well, what exactly do you know? If you don't even know how many reds in a triangle in Maxisnooker, how do you know that if you bring 9 sets along you will have exactly the right number of reds to form a triangle for us to play your Monstersnooker game?"

"I don't." replied Gordon. "In fact, I won't have the right number. So that's why I was going to ask you to bring some extra reds along, so we can form a triangle."

"But how many reds should I bring, if we don't know how many reds you will have with you?" asked Charlie.

"Well, Charlie," replied Gordon. "You're the clever one. Make sure that you bring enough extra red balls so that, no matter what the size of the triangles in Maxisnooker, we'll definitely have enough red balls, when you add yours to the 9 sets I am bringing, to form a perfect triangle."

"OK," said Charlie. "I'll do that. And I'll make sure I don't bring more reds than the maximum number that we might need."

"Good idea, Charlie." said Gordon. "No point in weighing yourself down with any balls that definitely won't be needed. If you've got any space left in your bag, we should fill it with bananas."


So, everyone.... how many red balls does Charlie bring that evening?

[snookersfun]

oh great , I knew I shouldn't have answered in my whoozy phase. At least I got rid of the 1/2 points that way. Good to be back.

[davis_greatest]

Congratulations snookersfun and davis_greatest!

Thank you for the point, davis_greatest, I'm glad I didn't break the table.

About Round 43, divide the point between snookersfun and yourself and let's move on!

Good idea! Half a point to each of us then!

The answer was Cantcope, with a chance of 7/11.

We know that the match went to a deciding frame, so it must have been 8-8 before the final frame. In coming back from 1-4 down to 8-8, Cantcope won 7 out of 11 frames. Therefore, the chance that he won frame 16 is 7/11 (or 63.63636363….%).

From 4-1 to 8-8, Robotsdaughter won 4 out off 11 frames, so the chance that he won frame 16 is 4/11.

We know immediately that Cantcope is more likely to have won frame 16, as he was behind, so needed to win more frames to level! (The 52% and 48% probabilities become irrelevant once we have the information that the match went to a deciding frame.)


SO HERE IS THE SCOREBOARD AFTER ROUND 46:

snookersfun……………………….…..19
abextra……………………………..…...10
Vidas……………………………………….8½
davis_greatest…………………..……5½
robert602…………………………………5
elvaago...............................4
Semih_Sayginer.....................½

(some rounds may be worth more than one point)
(especially ones won by davis_greatest)

[abextra]

Thank you for the point, davis_greatest, I'm glad I didn't break the table.

About Round 43, divide the point between snookersfun and yourself and let's move on!