Puzzles with numbers and things

[davis_greatest]

dantuck has asked to see the solutions to rounds 181 and 182, so here they are:

Round 181 solution

Frame score tied at 59-59 (I scored 59 in pots; Charlie 51 in pots + 8 in fouls) - then Charlie won 66-59 on respotted black.

I potted 7 reds, 2 yellows, 2 greens, 2 browns, 2 blues, 4 pinks.
Charlie potted 8 reds, 2 yellows, 1 green, 1 brown, 1 blue, 1 pink, 3 blacks and then a 4th black on the respot.

Round 182 solution

Frame score tied at 54-54 (I scored 54 in pots; Charlie 46 in pots + 8 in fouls) - then Charlie won 61-54 on respotted black.

I potted 7 reds, 2 yellows, 3 greens, 3 browns, 2 blues, 2 pinks.
Charlie potted 8 reds, 2 yellows, 1 green, 2 browns, 2 blues, 1 pink, 1 black and then a 2nd black on the respot.


Round 183 - still open!

Three correct solutions in so far - but I proved right when I said "needs a bit of care" ... anyone else who wants to answer, please do so on the thread, and see if you can be the first person to get it right first time with no slips!

[dantuck_7]

Round 184

Those puzzles stumped me! How about this for R184 - should be a bit simpler..

2,4,6,7,9, , , , , ? ,

What is the tenth number in this sequence.


Clue : Its not related to snooker - but similar...

[davis_greatest]

Rounds 181 and 182 are now closed. Please would someone post the answers on this thread.

Round 183 should be more straightforward. Just needs a bit of care...

Round 183 – Cheeky Charlie

I have just played a frame of spooker with my pet chimpanzee Charlie. Spooker has similarities with another, little-known, game called “snooker”, but with the following key differences.

In spooker,

- there is only one red ball;

- instead of there being six colours (worth from 2 to 7 points), there are 146 colours, worth from 2 to 147 points. They begin yellow (2 points), green (3), … black (7), just like snooker, but continue up to Silverback-gorilla-silver (146 points) and finally Orang-utan-orangey-gold (147 points);

- each time a ball (red or colour) is potted, it is replaced on the table;

- balls can be potted in any order, at any time, by either player – there is no need, for example, to pot a red before a colour;

- there are no fouls.

Each player must eat bananas and smoked salmon while playing. The frame finishes once either player has finished 17 bananas. At that time, whoever has scored more points wins.

The frame of spooker went, in many respects, like my frames of snooker in rounds 181 and 182. That is, by the time that Charlie finished his 17th banana (I was still only on my 9th):

I had scored more points from potting reds than Charlie had from potting yellows.
I had scored more points from potting yellows than Charlie had from potting greens.
I had scored more points from potting greens than Charlie had from potting browns...

... and so on, right up to...

I had scored more points from potting Silverback-gorilla-silvers than Charlie had from potting Orang-utan-orangey-golds.

We had also each potted each colour at least once.

Now, despite the above, Cheeky Charlie still managed to beat me! How, I don't know! I was so surprised by that that I examined things even more carefully, and I then discovered even more extraordinary facts:

- Given everything I have said above, it turned out that Charlie had potted the smallest number of reds possible.

- Given everything I have said above, it turned out that Charlie had potted the smallest number of balls possible!

How many reds did Charlie pot?

Who had potted more balls in the frame – and how many more had he potted than his opponent?

Answers initially by Private Message please...

Someone (either one of the three solvers, or anyone else who wishes to) please post the answer to R183.

I sent dantuck one possible answer to R184 - but it could be anything - so once the answer to R183 has appeared here I'll also move on with R185...

[snookersfun]

Answer to round 183: cheeky Charlie

Charlie potted 294 reds (making up for one extra point gained by d_g for every lower valued colour (yellow-146) and the 147.
He also potted 146 balls more alltogether (d_g having potted n balls of each value n, except one 147 and 3 reds vs n-1 balls of higher value of Charlie). Hope that sounds OK.

Roll on R185

[davis_greatest]

Round 183 - congratulations to snookersfun, Monique and abextra! Well done!

The fact that Charlie had potted the smallest number of reds possible makes it clear that:

- the number of points I scored from reds was only one point more than the number of points Charlie scored from yellows

- the number of points I scored from yellows was only one point more than the number of points Charlie scored from greens

... and so on, right up to...

- the number of points I scored from Silverback-gorilla-silvers was only one point more than the number of points Charlie scored from Orang-utan-orangey-golds.


When we also use the fact that Charlie had potted the smallest number of balls possible, we get a clear pattern –

I potted:

2 yellows (2 points each)
3 greens (3 points each)
4 browns (4 points each)
... and so on, right up to...
146 Silverback-gorilla-silvers (146 points each);

while Charlie potted

1 green (3 points each)
2 browns (4 points each)
3 blues (4 points each)
... and so on, right up to...
145 Orang-utan-orangey-golds (147 points each)

I also potted the minimum number (i.e. only one) Orang-utan-orangey-gold and Charlie potted the minimum number (i.e. only one) yellow (meaning I potted 3 reds).

For Charlie to win by one point, we soon get that he potted 2 x 147 = 294 reds, and potted 146 balls more than I.

Round 185 coming after lunch…

[davis_greatest]

Those puzzles stumped me! How about this for R184 - should be a bit simpler..

2,4,6,7,9, , , , , ? ,

What is the tenth number in this sequence.


Clue : Its not related to snooker - but similar...

Is it related to ten-pin bowling?

[davis_greatest]

Round 185 – Chirpy Charlie

I have just played yet another frame of lunchtime spooker with my pet chimpanzee Charlie. Charlie is being annoyingly chirpy because he beat me again - in fact, everything was exactly the same as round 183 above, right down to (and including) this line:

"- Given everything I had said above (in round 183), it turned out that Charlie had potted the smallest number of reds possible."

However, this time, Charlie had not potted the smallest number of balls possible, given everything above. In fact, he had potted more balls of one particular colour than in round 183 (but the same number of every other colour as he had potted in round 183).

In total, 101 more balls were potted than in round 183.

How many reds did Charlie pot this time?

Who had potted more balls in the frame – and how many more had he potted than his opponent?

What was the value of the particular colour which Charlie potted more often than he had done in round 183, and how many of that colour did he pot this time?

Answers initially by Private Message please...

[davis_greatest]

Round 185 is still open as some are still trying it. It has been successfully answered by snookersfun and abextra so far!

And for those who did round 185, a similar:

Round 186 – Chirpier Charlie

Charlie is even chirpier today, having beaten me once more at spooker - and, yet again, everything was exactly the same as round 183 above, right down to (and including) this line:

"- Given everything I had said above (in round 183), it turned out that Charlie had potted the smallest number of reds possible."

In total, however, 104 more balls were potted in the frame today than in round 183 and Charlie had potted more balls of one or more colours than in round 183 (and the same number of every other colour as in round 183). Interestingly, given everything above, the number of colours for which he had potted a different number of balls to round 183 was as small as possible! (I.e. For as many colours as possible, he had potted the same number of each as he had in round 183. )


How many reds did Charlie pot this time?

Who had potted more balls in the frame – and how many more had he potted than his opponent?

What was / were the value(s) of the colour(s) which Charlie potted more often than he had done in round 183, and how many more of each did he pot this time?

Answers initially by Private Message please...

[snookersfun]

I'll just add another easy one:

Round 187: Just a simple number

I am 5 digits long
I am divisible by 3 and 9, but not 6
My digits add up to 27
When my first and last digits are added, you will get a multiple of seven.
My lowest digit is 3
If you add up my second and fourth digits, you get 10
No number is used more than once
On the left of the comma, the numbers from left to right decrease by 1. On the right of the comma, the numbers increase by 3.

[snookersfun]

closing round 187:

the number is: 54,369

congratulations to abextra, dantuck, d_g, monique! (I didn't forget anybody, did I?)

[davis_greatest]

Round 185 – Chirpy Charlie

I have just played yet another frame of lunchtime spooker with my pet chimpanzee Charlie. Charlie is being annoyingly chirpy because he beat me again - in fact, everything was exactly the same as round 183 above, right down to (and including) this line:

"- Given everything I had said above (in round 183), it turned out that Charlie had potted the smallest number of reds possible."

However, this time, Charlie had not potted the smallest number of balls possible, given everything above. In fact, he had potted more balls of one particular colour than in round 183 (but the same number of every other colour as he had potted in round 183).

In total, 101 more balls were potted than in round 183.

How many reds did Charlie pot this time?[/B]

Who had potted more balls in the frame – and how many more had he potted than his opponent?

What was the value of the particular colour which Charlie potted more often than he had done in round 183, and how many of that colour did he pot this time?[/B]

And so it falls to me to close round 185! Congratulations to snookersfun, Monique and abextra for correct solutions! Well done!

And the answers:

How many reds did Charlie pot this time?

Well, since everything up to this point is the same as round 183, the answer to this must be the same! Still 294 reds, of course!

Who had potted more balls in the frame – and how many more had he potted than his opponent?

For Charlie to have potted the same (smallest possible) number of reds as before and still won, the value of the additional colours potted by Charlie must be the same as the value of the additional colours potted by me. He will then still win by one point.

As 101 additional balls were potted, if Charlie potted 50 extra colours each worth 51 points, and I potted 51 extra colours worth 50 points, they will be worth the same: 50 x 51 points!

So, Charlie still potted more balls - but instead of it being 146 balls more like in round 183, the difference it is now one fewer... 145 balls more.


What was the value of the particular colour which Charlie potted more often than he had done in round 183, and how many of that colour did he pot this time?

Answer is above - the value of the colour was 51 points. In round 183, he potted 49 of them - this time he potted 50 more, so 99 of them!

Easy, huh?!

[davis_greatest]

Round 186 – Chirpier Charlie

Charlie is even chirpier today, having beaten me once more at spooker - and, yet again, everything was exactly the same as round 183 above, right down to (and including) this line:

"- Given everything I had said above (in round 183), it turned out that Charlie had potted the smallest number of reds possible."

In total, however, 104 more balls were potted in the frame today than in round 183 and Charlie had potted more balls of one or more colours than in round 183 (and the same number of every other colour as in round 183). Interestingly, given everything above, the number of colours for which he had potted a different number of balls to round 183 was as small as possible! (I.e. For as many colours as possible, he had potted the same number of each as he had in round 183. )


How many reds did Charlie pot this time?

Who had potted more balls in the frame – and how many more had he potted than his opponent?

What was / were the value(s) of the colour(s) which Charlie potted more often than he had done in round 183, and how many more of each did he pot this time?

And also I must close round 186! Congratulations to snookersfun, Monique and abextra for correct solutions! Well done!

And the answers:

How many reds did Charlie pot this time?
Still 294, of course! Nothing has changed yet!

Who had potted more balls in the frame – and how many more had he potted than his opponent?
Charlie, 138 balls more - see below.

What was / were the value(s) of the colour(s) which Charlie potted more often than he had done in round 183, and how many more of each did he pot this time?
OK, 104 balls more were potted than in round 183. Don't fall for the trick of thinking that Charlie had to pot more than one colour differently from round 183 - he didn't! He still only needed to pot a different number of balls of one colour, when compared to round 183.

Suppose that, just as in round 185, I pot n+1 extra balls of colour valued n, and Charlie pots n extra balls of colour valued n+1.
That's 2n+1 extra balls potted, and my extra balls are worth the same as Charlie's - both n(n+1).

Now, you'll notice that 2n+1 is odd (and must be 3 or greater, as n>=1), but 104 is not odd! Well, that's no problem, because if for some k, I pot k(n+1) balls of value n, and Charlie pots kn balls of value n+1, they are still worth the same: kn(n+1) each.

So, keep dividing 104 by 2 until you get an odd number.
104, 52, 26, 13 - which is odd. So 2n+1 = 13, so n=6 (pink).
Now 104 = 8 x 13, so k=8.

Therefore, I potted an extra 8x7 = 56 pinks, and Charlie potted an extra 8x6 = 48 blacks.

[davis_greatest]

Round 188 - Potting mad!

I have spent all day trying to make a nice ring of snooker balls for The Statman's birthday celebrations. First, I put one ball down in the centre of my enormous snooker table and then made a single ring of balls around it, as close as possible to the centre ball. In fact, each ball in the first ring was touching the two adjacent balls in the ring, and also touching the centre ball - i.e. as closely packed as possible.

I had just got this far, when my silly little orang-utan Oliver potted all the balls I had just put down, so I had to start again! So, I laid out the balls as before, and placed another ring of balls around my first ring, again as close as possible to the centre. So, the new ring was one ball's width thicker than the first ring. But, again, would you believe it, Oliver potted all the balls, and I had to start again!

The 3rd time, my ring was one ball's width thicker than before, always as closely packed as possible - and just as I had finished that, Oliver potted all the #$&%*?£ balls!

And so it continued - each time that I had made my ring one ball's width thicker than I had managed the previous time, naughty Oliver potted all the balls and I had to start again! Over and over!

Oliver has just potted the balls 999 times. Nine hundred and ninety-nine times!!! But now, he has fallen asleep! So, I can finish The Statman's birthday surprise, and I get ready to start for the thousandth time.

How many balls has Oliver potted today?

(No calculators or computers allowed! "Experts" by PM initially please...)

[davis_greatest]

Correct answers to R188 so far from abextra, dantuck_7 and Monique .... well done. Round still open...

[davis_greatest]

Correct answers to R188 so far from abextra, dantuck_7 and Monique .... well done. Round still open...

Round 188 - Congratulations then to abextra, dantuck_7, Monique and now snookersfun for finding that Oliver had potted 999,999,999 balls. No wonder he became sleepy!

After n-1 times of potting the balls, Oliver had potted n³-1 balls.
Put n=1000, and you see that after 999 times, he had potted 1000³-1 = 999,999,999 balls.

The first time, he potted 7 balls (one central ball, with 6 balls around it);
the next time there were 19 balls (12 extra ones);
then 37 balls (18 extra ones)....
... each time the number of extra balls goes up by six, because the shape is hexagonal.

These numbers are the differences between successive cubes (this can be shown with algebra or geometrically):
2³-1³ = 8=1 = 7
3³-2³ = 27 -8 = 19
4³-3³ = 64 - 27 = 37
etc.

so adding them all up gives 1000³-1.