Puzzles with numbers and things

[dantuck_7]

While those are coming up, a quick one for those who dislike a bit of probability.

(b) (harder) When Nuggins plays Windywhirl, Nuggins’s superior skill makes a 4-2 final score (which could be to either ape) exactly 3 times as likely as a draw. What is the chance that they will share the first two frames?

Even though this is meant to be quickfire - can you give us all until midnight - have to get some work done!

[davis_greatest]

Even though this is meant to be quickfire - can you give us all until midnight - have to get some work done!

Then let's say any answers before 10a.m. BST tomorrow, send by PM please... and after that time, please put on the thread.

[davis_greatest]

One attempt in for round 254 (not quite right yet but close ).

Let me just clarify part (b)...

...(b) (harder) When Nuggins plays Windywhirl, Nuggins’s superior skill makes a 4-2 final score (which could be to either ape) exactly 3 times as likely as a draw. What is the chance that they will share the first two frames?

This is asking for the chance that the score will be 1-1 after the first 2 frames (not the chance that it will be 2-2 after 4 frames).

[davis_greatest]

Round 254 - Monique is in with the first correct answer to part (a)

[abextra]

Round 253 - Another Big Ape Break Bonanza

... identical to round 251, except that now potting a colour into a pocket of the same colour as the ball scores double points AND a penalty of 5 points (so potting pink into pink pocket would score 2x6 -5 = 7 points - that's double points for the pink but then 5 points deducted).

Break of 281:

purple in purple (17) - silver in green (9) - olive in olive (15) - orange in pink (8) -
purple in purple (17) - silver in green (9) - olive in olive (15) - orange in pink (8) -
purple in purple (17) - silver in silver (13) - black in black (9) - orange in orange (11) -
olive in olive (15) - silver in pink (9) - purple in purple (17)

2x(17+9+15+8)+17+13+9+11+15+9+17=189

15 reds

yellow in green (2) - green in yellow (3) - brown in blue (4) - blue in brown (5) - pink in pink (7) - black in black (9) - orange in orange (11) - silver in yellow (9) - olive in green (10) - purple in purple (17)

2+3+4+5+7+9+11+9+10+17=77

in total 189+15+77= 281

[Monique]

In this case ... 282 is possibe

yellow in green (2) - green in yellow (3) - brown in blue (4) - blue in brown (5) - pink in pink (7) - black in black (9) - orange in brown (8) - silver in silver (13) - olive in green (10) - purple in purple (17)

2+3+4+5+7+9+8+13+10+17=78

[davis_greatest]

In this case ... 282 is possibe

... silver in silver (13) - olive in green (10) ...

You cannot pot a ball into the green pocket immediately after one in the silver pocket, as they lie along the same edge.

281 is the highest possible.

[davis_greatest]

Round 254 - Monique is in with the first correct answer to part (a)

and dantuck_7 has given correct answers to rounds (a) and (b).

dantuck (or anyone else) - would you please put your explanations up on the thread? (I wasn't sure if your part (b) answer was a complete guess! )

[Monique]

my mistake ... yes you can't pot olive anywhre if you want purple in purple.

[dantuck_7]

R254

a). There are 30 combinations that would result in a scoreline of 4-2 or 2-4 out of 2^6 possible combinations.

So thats 30/64.

Part b) First work out the probability of Nuggins to win a single frame (came to 0.211.....) then work out the chance of 1-1 after 2 frames.

Wil expand on this later unless anyone else wants to post. Monique?

[davis_greatest]

R254

a). There are 30 combinations that would result in a scoreline of 4-2 or 2-4 out of 2^6 possible combinations.

So thats 30/64.

Part b) First work out the probability of Nuggins to win a single frame (came to 0.211.....) then work out the chance of 1-1 after 2 frames.

Wil expand on this later unless anyone else wants to post. Monique?

Congratulations, dantuck, I'm sure it wasn't a guess then! A couple of points though:

- you don't need to work out the probability of Nuggins winning a single frame, since as you have seen, that is rather a messy number - you can get to the answer more directly;

- if you do go down that route, the probability you have given is that of Windywhirl winning a frame - remember that Nuggins is better than Windywhirl so Nuggins's chance of winning each frame is greater than 1/2.


PS If I get a minute, I'll edit this post and put the solution to part (b) in hidden text...


Edit: here is explanation for part (b) in hidden text - select to read.
(
The answer to part (b) is 1/3.

We can see this as follows:


Let
N = chance that Nuggins wins each frame
W = chance that Windywhirl wins each frame

So .............(i) N+W = 1

Square equation (i) to get
............(ii) (N+W)² = N² + W² + 2NW = 1

Number of ways of choosing 2 frames from 6 for a player to lose is (6 x 5) / (2 x 1 ) = 15

So
chance that Nuggins wins 4-2 is 15 N^4 W²
chance that Windywhirl wins 4-2 is 15 N² W^4

So chance of 4-2 result is:

............(iii) 15 (N^4 W² + N² W^4)

Number of ways of choosing 3 frames from 6 is (6 x 5 x 4) / (3x2 x 1 ) = 20

So chance of 3-3 draw is
............(iv) 20 N³ W³

We were told that (iii) = 3 x (iv), i.e.

15 (N^4 W² + N² W^4) = 60 N³ W³

Divide both sides by 15 W² N² to get

N² + W² = 4 NW
or
............(v) N² + W² + 2NW = 6 NW

Substitute (ii) into (v) to get 1 = 6NW

Chance of the first 2 frames being shared is NW + WN = 2NW = 1/3.

)

[davis_greatest]

OK, a very easy one – with no probability!

Round 255 (=1+2+4+8+16+32+64+128) – Snooker Triangle

Set up the snooker balls as usual for the start of a frame (ignore the cue ball – leave it in your pocket / handbag). Now, move as few balls as possible in order to rearrange the balls into one equilateral triangle, with every horizontal row* worth the same number of points. The triangle must have its apex at the same place as it would usually be at the start of a frame, i.e. as close as possible to (without touching) a hypothetical ball on the pink spot.

*a horizontal row means a row parallel to the top cushion

How many balls do you move, and where?

Answers by Private Message / pager please

[davis_greatest]

Round 255 update - Monique has now answered this... and snookersfun also has the arrangement and just needs to count how many balls were moved, in order to answer the question!

[snookersfun]

R.254: either this or that...

[ATTACH]978[/ATTACH]

same as usual, just two clue options for each line, so you have to choose the correct clue. Have fun

update: have 2 correct answers for the last battleship by Monique and Abextra so far. Well done
The next answer can go onto the thread, after tomorrow 7pm BST Monique or Abextra can put up their answers.

... and snookersfun also has the arrangement and just needs to count how many balls were moved, in order to answer the question!

uproarious!!!!

[davis_greatest]

OK, a very easy one – with no probability!

Round 255 (=1+2+4+8+16+32+64+128) – Snooker Triangle

Set up the snooker balls as usual for the start of a frame (ignore the cue ball – leave it in your pocket / handbag). Now, move as few balls as possible in order to rearrange the balls into one equilateral triangle, with every horizontal row* worth the same number of points. The triangle must have its apex at the same place as it would usually be at the start of a frame, i.e. as close as possible to (without touching) a hypothetical ball on the pink spot.

*a horizontal row means a row parallel to the top cushion

How many balls do you move, and where?

Answers by Private Message / pager please

... correctly answered by Monique, snookersfun and abextra. Can we have the answer on the thread please...