Puzzles with numbers and things

[snookersfun]

Addendum R. 361

after having enlisted the help of the world's packing-of-circles-in-square experts with all their computational power, we can now be confident that 316 balls or more indeed do not fit into a 17x17 box.
Here is the list I obtained (thanks to Dr. Eckard Specht), also whoever still interested look at his site and be amazed by the irregular arrangements of most packings and play around with their cool ball/box calculator
http://www.packomania.com/

N___radius (for box size of 1)___box size (for diameter of 1)
----------------------------------------------------------------
310 0.029656532915.................16.859
311 0.029583286119.................16.901
312 0.029572068491.................16.908
313 0.029568593144.................16.910
314 0.029530610813.................16.932
315 0.029408808740.................17.002
316 0.029319864182.................17.053
317 0.029276000746.................17.079
318 0.029213599497.................17.115
319 0.029173220387.................17.139

slight aside, erm, actually, they don't even fit the obvious 315 balls inside the 17x17 box
But that just must be due to the fact that they approach the problem the other way around, i.e. find the smallest possible square box for a given number of balls or rather they calculate the radius of the given number of balls fitting into a unit box.

and last, but not least, Moglet improved on my wobbly row (involving the outside balls of the next row above the wobbles), as there was no way to fit the balls in the way I thought (deluded by silly paint).

...and now I have to go find a life again

[davis_greatest]

Round 362 - Chess with balls

Gordon is playing snooker against Charlie, while Oliver, who has spent the whole weekend playing chess, referees.

Gordon makes a total clearance, while Oliver proceeds to record it by colouring in his old 6x6 grid scoreboard. When Gordon pots his first red, Oliver colours in a square red. For each following shot, Oliver proceeds to move around the board 2 squares in one direction and one square the other (at right angles) - rather like a knight moves in chess, colouring the square in the same colour as the ball potted.

Once Oliver has coloured in the final black, he is able (through his clever planning) to make one move again in the same way (like a knight) and be back at the opening red square again! Below is his completed scoreboard.

Find the opening red square and show the order in which Oliver recorded the squares.


Answers by private message initially please...

[davis_greatest]

Thanks Monique. The weakness in this, however, is that it unfortunately assumes that all balls must be placed in straight rows, which is not necessarily the case. We need also to consider the combinations of wobbly rows (or other arrangements) that could be produced!

[Monique]

Well I think here is a proof that indeed 315 is the maximum.
Obviously in a single row we can have a maximum of 17 balls. If all rows are of 17 balls we can only have 17 rows, center of balls being aligned parallelly to the sides of the box. "Shifting" the balls in one row of of two allows to get squeeze more rows in the box at the price of having one row out of 2 featuring 16 balls instead of 17, because a shifted row can only fit 16 balls in the 17 size box. The maximum number of rows being achieved if the balls are organised at 60° angles like in the snooker triangle. For such arrangement the total "lenght" of a set of n rows is 1+(n-1)*sqrt(3)/2.
We all found out that n can be maximum 19 if the size of the box is 17. With all rows in 60° pattern we have 314 balls and some space left. To get 315 balls we must introduce one more row of 17 balls, so mixing 60° pattern - in 16 rows - and 90° pattern - in 3 rows. For 316 balls we would need a 14 rows 60° pattern and 5 rows 90° pattern...

Now if k is the maximum number of rows in 90° pattern we can achieve in the 17 size box, we must have k the biggest integer such as k+1 <= (17-19*SQRT(3)/2)/(1-SQRT(3)/2) which yields k=3 as a maximum. Hence 315 balls is the maximum.

Hope this makes sense ...

[davis_greatest]

Thank you also snookersfun and abextra for the pictures for 315 balls! Showing that there are some very different ways of getting this.

Given the (infinite) number of possible arrangements to make use of the gaps, it is far from easy here to prove what is the maximum possible (other than showing that 333 must be an upper bound) - although I suspect that 315 cannot be improved upon for the dimensions of Charlie's box.

[PaulTheSoave]

I dont mind, I am not the quiz master.

Thank you, Paul! Tried to make the counting easier...

[abextra]

But cute colours.

Thank you, Paul! Tried to make the counting easier...

[PaulTheSoave]

But cute colours.

Wow, Moglet, you are rich!!!

This is my pic - not as golden...

[abextra]

Oh dear!

[ATTACH]1907[/ATTACH]

Wow, Moglet, you are rich!!!

This is my pic - not as golden...

[snookersfun]

well, here is my wobbly one, is only in scrap-book quality, but goes up as is now, might be worked over later on, but very possibly not, as can't beat moglet's coins anyway

balls-4.bmp
left side, the box filled neatly in rows and columns
right side:
9 rows of 17 balls each
9 rows of 16 balls each
last wobbled row with 18 balls (those red ones, ignore all other circles there)

[moglet]

Very nice moglet! 315 indeed! (And yes, the geometry does work - you would have another 0.96% width of a snooker ball spare at the top under this arrangement.) Just where did you get all those coins? Now you have to show it with snooker balls

There are other arrangements possible - any others to go up on the thread?

I always knew they would come in useful one day

[davis_greatest]

Very nice moglet! 315 indeed! (And yes, the geometry does work - you would have another 0.96% width of a snooker ball spare at the top under this arrangement.) Just where did you get all those coins? Now you have to show it with snooker balls

There are other arrangements possible - any others to go up on the thread?

[Monique]

Oh dear!

[ATTACH]1907[/ATTACH]

Absolutely brilliant moglet ...

[moglet]

Such was the confusion in this round, I believed I had to make more than 315, so, geometry failed me, to find more I tried it practically

[snookersfun]

[ATTACH]1907[/ATTACH]

brilliant! Now that is a proper proof!
and I do see 2 solutions above there, but showing each only part of the box...