Puzzles with numbers and things

[dantuck_7]

Perfect answers to rounds 181 and 182 have been received from abextra, snookersfun and Monique. If anyone else wants a go, please answer on the thread. I'll leave both rounds open until some point tomorrow, when I'll hopefully then post round 183.

If someone could post the answers to 181 or 182 - it would be good. Had no luck with this one - Found a score of 50-50 but then the number of reds potted by DG was equal to the number of points scored in yellows by the chimp.

[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...

[davis_greatest]

Rounds 181 and 182 are both still open...

Perfect answers to rounds 181 and 182 have been received from abextra, snookersfun and Monique. If anyone else wants a go, please answer on the thread. I'll leave both rounds open until some point tomorrow, when I'll hopefully then post round 183.

[davis_greatest]

Round 181 - Clever Charlie is, of course, still open! However, for anyone finding it too easy (it didn't seem to take abextra long at all!), you can also try Round 182, which is exactly the same, except for the last bit in green.

Round 182 - Cleverer Charlie

I’ve just finished another fascinating frame of snooker against Charlie. At the start of the frame, things didn’t go so well for me, when I gave away two 4-point fouls in failing to get out of a snooker.

However, those proved to be the only fouls of the frame; and after that bumpy start, things went better. In fact, I seemed to be outshining Charlie in every department, until he potted the final black to tie the frame – in fact, once he had potted it, I noticed the following extraordinary facts:

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.
I had scored more points from potting browns than Charlie had from potting blues.
I had scored more points from potting blues than Charlie had from potting pinks… and
I had even scored more points from potting pinks than Charlie had from potting blacks!

Charlie had also potted each colour at least once.

There were no free balls in the frame, never was more than one ball potted in one stroke, and a colour was potted immediately after each of the 15 reds (i.e. 15 reds and 15 colours were potted, before getting to the final colours).

Unfortunately, after looking at all those statistics, it turned out to be for nothing. For Charlie then potted the respotted black to steal the frame - and when he had done so, it was the first time during the frame that our combined scores from pots (excluding the 8 penalty points) had totalled more than 100 points!

What was the final score and how many reds and each colour did we each pot?

"Experts'" answers by PM, initially please...

So, snookersfun is first there with a correct solution to round 182. Congratulations!

Rounds 181 and 182 are both still open...

[davis_greatest]

Monique has joined abextra in having answered round 181!

And so has snookersfun!

snookersfun has also discovered that round 182 as originally incorrectly worded is impossible... the wording is now corrected above in the final bit in bold green. Ooops... sorry!

[davis_greatest]

Monique has joined abextra in having answered round 181!

[davis_greatest]

Round 181 - Clever Charlie is, of course, still open! However, for anyone finding it too easy (it didn't seem to take abextra long at all!), you can also try Round 182, which is exactly the same, except for the last bit in green.

Round 182 - Cleverer Charlie

I’ve just finished another fascinating frame of snooker against Charlie. At the start of the frame, things didn’t go so well for me, when I gave away two 4-point fouls in failing to get out of a snooker.

However, those proved to be the only fouls of the frame; and after that bumpy start, things went better. In fact, I seemed to be outshining Charlie in every department, until he potted the final black to tie the frame – in fact, once he had potted it, I noticed the following extraordinary facts:

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.
I had scored more points from potting browns than Charlie had from potting blues.
I had scored more points from potting blues than Charlie had from potting pinks… and
I had even scored more points from potting pinks than Charlie had from potting blacks!

Charlie had also potted each colour at least once.

There were no free balls in the frame, never was more than one ball potted in one stroke, and a colour was potted immediately after each of the 15 reds (i.e. 15 reds and 15 colours were potted, before getting to the final colours).

Unfortunately, after looking at all those statistics, it turned out to be for nothing. For Charlie then potted the respotted black to steal the frame - and when he had done so, it was the first time during the frame that our combined scores from pots (excluding the 8 penalty points) had totalled more than 100 points!

What was the final score and how many reds and each colour did we each pot?

"Experts'" answers by PM, initially please...

[davis_greatest]

I have just edited the question in round 181 above, as I had originally forgotten to include the bit I have now added in italics.

I see that abextra has already solved it, anway. Well done!

"Experts'" answers by PM, initially please...

[davis_greatest]

Round 181 - Clever Charlie

Let’s see how you get on with this one!

I’ve just finished another fascinating frame of snooker against Charlie. At the start of the frame, things didn’t go so well for me, when I gave away two 4-point fouls in failing to get out of a snooker.

However, those proved to be the only fouls of the frame; and after that bumpy start, things went better. In fact, I seemed to be outshining Charlie in every department, until he potted the final black to tie the frame – in fact, once he had potted it, I noticed the following extraordinary facts:

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.
I had scored more points from potting browns than Charlie had from potting blues.
I had scored more points from potting blues than Charlie had from potting pinks… and
I had even scored more points from potting pinks than Charlie had from potting blacks!

Charlie had also potted each colour at least once.

There were no free balls in the frame, never was more than one ball potted in one stroke, and a colour was potted immediately after each of the 15 reds (i.e. 15 reds and 15 colours were potted, before getting to the final colours).

Unfortunately, after looking at all those statistics, it turned out to be for nothing. For Charlie then potted the respotted black – the fourth time that he had potted the black in the frame – to steal frame and match!

What was the final score (and, if you can manage it, how many reds and how many of each colour ball did we each pot)?

[Monique]

R180 ... is closed

Once again well done to D_G, Snookersfun and Abextra!

[snookersfun]

so, only to add the shortcut way to the perfect solutions above:
Another way to look at it is to imagine that the brown tetrahedron has n balls along each edge, and 3 faces of dirty balls. Going to the yellow tetrahedron we added in one more outer row of dirty balls at the bottom of the pyramid, made up of (n+1)+n+(n-1) = 3n balls. Therefore 45/3=n. n (base) of the smaller tetrahedron = 15.

[snookersfun]

I think the big cube was made of 125 smaller cubes (5*5*5) and there were 4 sides painted grey (let's say, all four walls were painted grey, ceiling and floor remained magenta)?

perfect Abextra! I see you are working on your backlog

[abextra]

Round 179 - From spherical elephants to cubic ones

Imagine a big magenta cube built from many smaller unit cubes (e.g. produced from shaping some magenta coloured elephants into little unit cubes). Now imagine somebody painting some sides of the big cube 'grown-up'elephant grey. Given that after taking that big cube apart you find 45 elephants still all over magenta, as before, how many elephants made up that big cube and how many sides of it were painted?

I think the big cube was made of 125 smaller cubes (5*5*5) and there were 4 sides painted grey (let's say, all four walls were painted grey, ceiling and floor remained magenta)?

[abextra]

Another way to look at it is to imagine that the yellow tetrahedron has n balls along each edge, and 3 faces of dirty balls. Each time you remove one face of dirty balls, you are left with a smaller tetrahedron.

After removing 3 faces of dirty balls, you have a tetrahedron of size n-3, and have removed triangles of length n, n-1 and n-2;

i.e. there are

T(n) + T(n-1) + T(n-2) dirty yellow balls, (1)

where T(n) = the n-th triangular number = n(n+1)/2

The brown tetrahedron has n-1 balls along each edge, so there are
T(n-1) + T(n-2) + T(n-3) dirty brown balls (2)

We know that (1) - (2) = 45,
i.e. T(n) - T(n-3) = 45.

This can be rearranged as 3(n-1) = 45, so n=16.

Then plug n=16 into (1) to get 361 minutes!

Very nice, d_g, must be easy to live with your brains...

[davis_greatest]

Another way to look at it is to imagine that the yellow tetrahedron has n balls along each edge, and 3 faces of dirty balls. Each time you remove one face of dirty balls, you are left with a smaller tetrahedron.

After removing 3 faces of dirty balls, you have a tetrahedron of size n-3, and have removed triangles of length n, n-1 and n-2;

i.e. there are

T(n) + T(n-1) + T(n-2) dirty yellow balls, (1)

where T(n) = the n-th triangular number = n(n+1)/2

The brown tetrahedron has n-1 balls along each edge, so there are
T(n-1) + T(n-2) + T(n-3) dirty brown balls (2)

We know that (1) - (2) = 45,
i.e. T(n) - T(n-3) = 45.

This can be rearranged as 3(n-1) = 45, so n=16.

Then plug n=16 into (1) to get 361 minutes!