Puzzles with numbers and things

[Vidas]

tiling(question 25)...

147x147, total 21609 squares.

Let's paint bathroom floor, using 4 colors(red, blue, green, white)

rbrbrbrbrbrb...r
wgwgwgwgwgwg...w
rbrbrbrbrbrb...r
wgwgwgwgwgwg...w
.....
rbrbrbrbrbrb...r

red/blue/green/white squares:
red = 74*74 = 5476
blue = 73*74 = 5402
white = 73*74 = 5402
green = 73*73 = 5329.

any 4-inch piece covers red,blue,white, and green squares.(a - amount of 4-pcs.)
3-inch piece can be of 4 kinds: b(covers red, white, blue), c(red, green, blue), d(red,white, green), e(white, green, blue).
a+b+c+d=5476
a+b+d+e=5402
a+b+c+e=5402 so d=c
a+c+d+e=5329.
-------------------
as d=c,
a+b+2c=5476
a+b+c+e=5402
a+2c+e=5329.
(1)-(2): c-e=74,-> c>=74. -> a<=5329-2*74 =5181, 4a<20724
So b+c+d+e >=(21609-20724)/3=295. total amount of 3-inch pieces exceeds 295.
295=147*2+1, so WT will use at least one colour more than twice.

[davis_greatest]

In simple terms, you need to find the SMALLEST number of colours to make the game possible - and prove that it is the smallest.

So, if for example you find a number of colours that works with 1,000 balls - as you have done - you must show that no one could ever find a smaller number of colours that would work with any other number of balls (10000, 100000, or 1000........ with any number of zeroes).

[davis_greatest]

Not sure, if I got the whole story, and don’t have time to double check. So I just throw this in:
How about the white, a pile of 900 (30x30), one of 81, and two of 9 each.

Sounds to easy somehow, will definitely check back later.

There are essentially two things that you need to show:

1) You need to find a number k of colours and show that it IS possible to satisfy the rules of the game using these k colours.

2) You need to show that it would NOT be possible to satisfy the rules of the game using fewer than k colours.


So, you have done part 1) - i.e. you have shown that with k=5 colours and 1000 balls, it is possible to play.

But if you think that 5 colours is the fewest possible, you need also to show part 2) - i.e. show that with fewer than 5 colours (i.e. 2, 3, or 4 colours) it would be IMPOSSIBLE to have exactly 10,000 balls, or 100,000, or 1,000,000, or 10,000,000 or indeed ANY bigger number at all that begins 1 followed by only zeroes.

[snookersfun]

Not sure, if I got the whole story, and don’t have time to double check. So I just throw this in:
How about the white, a pile of 900 (30x30), one of 81, and two of 9 each.

Sounds to easy somehow, will definitely check back later.

[davis_greatest]

Question 26 - Squaresnook

Remember that questions 25 and 25½ are still live, but here is question 26 if you are struggling with those.

While on holiday last week, having finished writing my latest book "Enhance Your Memory and Never Forget A Thing... Ever and Enhance Your Memory" I had the pleasure of seeing in a distant land a game of Squaresnook. I even had a go. It is played on an enormous billiard table, with very, very many balls, and here is how the balls are set up:

Rule 1) There are balls of different colours. [I don't remember how many colours - all that I remember is that there were at least two colours.]

Rule 2) One of the colours is white. There is only one white ball and it is used to hit the other balls.

Rule 3) For each colour other than white, there is one or more balls of that colour. Balls of the same colour are arranged in a square formation. Squares (if there is more than square) need not be the same size.

[As I said, I don't remember how many colours there were - only that there was at least one colour other than white. For instance, there could have been a square of pink balls (e.g. just 1 pink ball, or perhaps 2x2 = 4 pink balls, or 1000x1000 = 1 million pink balls, or whatever), and another square of gold balls, or whatever colours there were.]

Rule 4) The total number of balls must be at least 1,000 and must be exactly 1 followed by some zeroes.

[There were perhaps 1,000 balls on the table altogether, or 10,000, or 100,000 - I don't remember. Maybe 10000000000000000000 or more, for all I can remember. It's all a distant blur.]



While I was watching, I remember thinking to myself that they were using the smallest number of colours possible in order to play by the rules.

How many colours did they use in the game of Squaresnook I watched? (and you must prove you are right!)

[Obligation]

No, but I recently went to an Accountants' Convention where they wanted to prove that accountants are not stupid. So they called an accountant up on stage and the host asked him "What is 19 plus 19?"

"6.3" replied the accountant. All the accountants at the convention sighed in disappointment, but as the accountant was about to go off, they started chanting "Give him another chance! Give him another chance!"

"OK," said the host. "One more chance. What is 11 plus 11?"

The accountant paused for a moment. Then he declared loudly "283." There was a big disappointed silence, as the accountants realised that they had failed to prove that they are not stupid. But as the accountant was about to leave in shame, the audience started chanting again "Give him another chance! Give him another chance!"

"OK, last chance," said the host. "What is 2 plus 2?"

The accountant on stage stopped for a few seconds and opened his mouth as if to answer. There was a deep hush and he stopped and thought some more. Finally, he declared, with a confident bellow "Two plus two is four!"

There was a huge silence. Then, up came the roar from the audience: "Give him another chance! Give him another chance!"

Lol

[davis_greatest]

Welcome back, davis_greatest. glad you had a good time.

Thank you, Vidas. Certainly did!

[davis_greatest]

DGE are you an accountant?

No, but I recently went to an Accountants' Convention where they wanted to prove that accountants are not stupid. So they called an accountant up on stage and the host asked him "What is 19 plus 19?"

"6.3" replied the accountant. All the accountants at the convention sighed in disappointment, but as the accountant was about to go off, they started chanting "Give him another chance! Give him another chance!"

"OK," said the host. "One more chance. What is 11 plus 11?"

The accountant paused for a moment. Then he declared loudly "283." There was a big disappointed silence, as the accountants realised that they had failed to prove that they are not stupid. But as the accountant was about to leave in shame, the audience started chanting again "Give him another chance! Give him another chance!"

"OK, last chance," said the host. "What is 2 plus 2?"

The accountant on stage stopped for a few seconds and opened his mouth as if to answer. There was a deep hush and he stopped and thought some more. Finally, he declared, with a confident bellow "Two plus two is four!"

There was a huge silence. Then, up came the roar from the audience: "Give him another chance! Give him another chance!"

[davis_greatest]

Round 25½

How many and which European countries did I see on my holiday?

[Vidas]

Welcome back, davis_greatest. glad you had a good time.

[Obligation]

DGE are you an accountant?

[April madness]

Hi. Really good holiday, thanks. Saw quite a few European countries.

Welcome back!

Now for a change you could ask a question about how many and which European countries did you see on your holiday

[davis_greatest]

Hi. Really good holiday, thanks. Saw quite a few European countries.

If you want to see that the number of possible lists with 5 breaks (each of which can take 80 possible values from 68 to 147) is the same as the number of ways of choosing 5 objects from 80+5-1 objects, just observe that each list corresponds to choosing how many times each of the 80 possible breaks can appear, subject to the total number of times adding up to 5.

As before, use matchsticks and lemons to represent each possible list of 5 breaks.

First, put down a number of lemons equal to the number of times the lowest possible break (68) appears on a list. Then put one matchstick. Then put down a number of lemons equal to the number of times a 69 break appears and then put one matchstick.

Then put down a number of lemons equal to the number of times a 70 break appears, then one matchstick, .... continue, until you put down a number of lemons equal to the number of times a 146 break appears, then one matchstick. Finally put down a number of lemons equal to the number of times a 147 break appears.

We need to use 5 lemons (as there are 5 breaks on the list) and 79 separating matchsticks. So there are 84 objects, and the number of lists is the same as the number of ways of making 5 of the objects be lemons and the rest matchsticks.



HERE IS THE SCOREBOARD AFTER ROUND 24

snookersfun……………………….…..12
Vidas……………………………………….6½
robert602…………………………………4
abextra……………………………..…...3½

(some rounds may be worth more than one point)


Yes, round 25 with WT's bathroom tiling is still open. I don't think I've seen any attempts yet! Here it is:
http://www.thesnookerforum.com/showt...2-page_28.html

[snookersfun]

Good one! You could have got there a bit faster by noting that the number of different lists (in descending order) of 5 breaks, each of which can take 80 different values, is the same as the number of ways of choosing 5 objects from 80+5-1=84 objects.

I.e. it is 84! / 5!(84-5)! = 30,872,016. So, with one bag being given every million days, there are 30 bags.

Definitely 2 points for this!

Well, thank you! And trust me, I wouldn't have... noted. At least, I noticed it is not 80x80x80x80x80/5!

So, how was the vacation? Really missed you on here, although I wouldn't have had time for math anway.
You noticed we didn't tile either...

[davis_greatest]

It's worth lots of points. Let me find my scribbles.
Did you see the welcome-back post above. You seem to be deeply absorbed in the math!

OK, found something. It's messy, trying to extract:
I added up all the possible combinations:
1) w.o. repeated numbers (there are 80 possible different high-breaks possible): 80!/5!(80-5)! = 24.040.016
2) all #s the same = 80
3) 4 of the same #s = 80x79 = 6320
4) 3 of the same #s + 2 of same #s = 80x79 = 6320
5) 3 of same + 2 different #s = 80x79x78/2 = 246.480
6) 2 of same + 3 singles = 80x79!/3!(79-3)! = 6.326.320
7) 2 pairs + single like 5

sum was less than 31.000.000, therefore 30 bags

Good one! You could have got there a bit faster by noting that the number of different lists (in descending order) of 5 breaks, each of which can take 80 different values, is the same as the number of ways of choosing 5 objects from 80+5-1=84 objects.

I.e. it is 84! / 5!(84-5)! = 30,872,016. So, with one bag being given every million days, there are 30 bags.

Definitely 2 points for this!