Puzzles with numbers and things

[davis_greatest]

LOL - that's more like it! Great elephants!

[dantuck_7]

http://www.thesnookerforum.com/photo....php?n=348&w=l


Here is the pentagon of mine for 15 pts. I think this is a bit nicer than those triangles. Half a point for a single line connecting 2 vertex's would have added an extra 2.5 points. If it had been 11 elephants you can see from the dotted line I added in that would have made 19 pts.

[davis_greatest]

http://www.thesnookerforum.com/photo....php?n=348&w=l


Here is the pentagon of mine for 15 pts. I think this is a bit nicer than those triangles. Half a point for a single line connecting 2 vertex's would have added an extra 2.5 points. If it had been 11 elephants you can see from the dotted line I added in that would have made 19 pts.

Yes, I quite like the pentagon - although of course there are no points awarded for joining just 2 elephants so you can't have the outside lines. This then becomes a star, the same as snookersfun's star above...

[snookersfun]

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?


expert answers by PM.

[snookersfun]

OK, expert answers are in and good as usual. Well done d_g and Monique!
Anyone else?

[Monique]

Round 180 - An injustice...

This morning I went to Barry the Baboon's Ball Shop. As I entered the shop I immediately noticed that the atmosphere was tense and that Barry looked exhausted. The shop was packed – I spotted Charlie, Gordon, Oliver, Rollie O'Sunnyman and Damon Grott – and Barry was talking excitedly.
"Yesterday morning", he said "I decided to redecorate my shop window. I built a nice regular triangular pyramid with yellow balls I had in stock. The basis of this pyramid is an equilateral triangle; you know the shape of the racks used for orang-utan snooker also played by humans". "So" said Oliver "You built a regular tetrahedron". "Wha?" asked Rollie, "What?" echoed Grott. "A tetrahedron" repeated Oliver … nobody argued further.
"Never mind!" said Barry, "In the afternoon while I was busy with customers, some idiot poured chocolate sauce on my nice theaterdon! All visible balls were coated with it and sticky. Last night I spent an awful lot of time cleaning them. Mind you, I needed one full minute for each ball. It's a shame". "Indeed" murmured a dreamy Rollie "All that delicious chocolate…" Barry glared at him but said nothing (Rollie is a very good customer…)
"This morning, before opening the shop, I built another teadragon, this time with brown balls. It isn't quite as big as the previous one. In fact its edges are one ball shorter than those of the yellow one. And now look at this". Barry was pointing to a glossy dark brown mass, in a dramatic gesture. "Humm, caramel topping" said Rollie, licking a finger "just like bresilienne ice-cream". Barry's face turned dark purple, but he still kept silent.
"I will have to clean all this mess; another sleepless night probably …" sighted Barry. "Comooon! At least it will take you three quarter of an hour less than yesterday" said Charlie, who was standing behind Grott, in a high pitched squeaky voice.
That was the last straw for Barry. He jumped at poor Grott and slapped him hard in the face repeatedly! And nobody argued … and nobody counted the slaps.

How long did Barry spend cleaning the yellow balls last night?

Expert answers by PM...

[Monique]

Snookersfun was first to answer (right of course!). Well done

[Monique]

And D_G also. Congratulations!
It remains open ...

[abextra]

This morning I went to Barry the Baboon's Ball Shop....

How long did Barry spend cleaning the yellow balls last night?

My first guess - 6 hours and 1 minute?

[Monique]

Yes! That's it. Well done abextra. Would you please explain your solution on the thread?

[snookersfun]

My first guess - 6 hours and 1 minute?

Heyhey! Abextra is back and puzzling!!! Welcome back!

[t.lavery55]

8 hours and 3 minutes?

[davis_greatest]

Hehe - welcome back, abextra!

[abextra]

Thank you, Snookersfun and davis_greatest!

Yes! That's it. Well done abextra. Would you please explain your solution on the thread?

Oh, the explaining is much harder than solving!

Ok, let's say Barry the Baboon has a tetrahedron with n balls on every edge. Three sides of this tetrahedron are visible, it means he has to clean only the balls which are on these three sides.

On the first side there are n(n+1)/2 dirty balls.

On the second side, the balls on one edge are already cleaned, so there are n(n+1)/2 -n dirty balls.

On the third side the balls on two edges are cleaned, so there are n(n+1)/2 -2n+1 dirty balls.

If we add up all the dirty balls, we see that in total Barry the Baboon has to clean 3n(n-1)/2 +1 balls.

Now, let y be the number of yellow balls on an edge and let b be the number of brown balls on an egde. Then

3y(y-1)/2 +1 - 3b(b-1)/2 +1 = 45.

The number of brown balls on the edge was 1 less than the number of yellow balls on the edge, so we can replace b by y-1 and after some calculations we get that y=16, i.e.

there were 16 balls on an edge of the tetrahedron of yellow balls and on the three visible sides there were 361 balls.

As it took 1 minute to clean one ball, Barry the Baboon had to waste 361 minutes or 6 hours and 1 minute to clean all the dirty yellow balls. BTW, it will take 5 hours and 16 minutes to clean the dirty brown balls... of course he was angry!

Sorry for my English!

[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!