I think the diagram above illustrates the (5x5) + (5x4) solution that you posted above. The top diagram is a “view from above” whilst the lower diagram is a typical “x-section”. Both showing the first two layers in each case.
Hopefully it helps illustrate how the 0.866 “layer thickness” is derived (ie sqrt 0.75)
If I have mis-understood your proposed solution just let me know and I’ll have another shot at the calculation and illustration (I just use Powerpoint for these diagrams)
Two railway trains, one 400ft long, the other 200 ft long, run on parallel tracks.
When they run in opposite directions, they pass each other in 5 seconds.
When they run in the same direction, the faster train takes 15 seconds to overtake the slower train.
What is the speed of each train ?
Note
each train runs at a constant speed throughout.
You can quote the speeds in ft/sec, but
Just for fun, more marks if you quote the speeds in mph using vulgar fractions
Yes, it seems that calculating that one I omitted the final sq rt. So I am going to stick with my answer in post 597, the second arrangemey fitting 272 balls, even if there seems to be a lot of free air… In my head I cannot envisage any arrangement squeezing more in.
I’ll leave it open for a few days to let others see if they can figure out how to get 288.
Tantalizing, once the concept has been grasped, the “arithmetic” involves nothing more complicated than a “3/4/5” Pythagorean triangle. In other words, the proof can be done mentally, or with a scrap of paper and pencil if really pushed
I’ve been thinking about those balls and suspect the answer lies in not having the bottom layer filling the entire space, which the next layer would. But I’m too tired to solve it.
The Brain Teaser thread is meant to be a haven for relaxation Mike !!
I’ll post my solution when I get home tonight.
Meanwhile, don’t have (Climate Change) nightmares, and i’m sure that if I mention that some of the multiples of 288 include 12 x24 and 16 x 18 you will drift into a deep, deep slumber in no time at all
Hi Mike, Hi IB and any others who might still be remotely interested !
Well, its night again and i’m back from work.
The 16 x 18 multiple of 288 were obviously a hint.
However, firstly, think of the 5x5x10 box as an "upright box with a square base 5x5 and a height of 10 - a bit like a box containing a bottle of whisky
The diagram below shows how to lay out 18 balls and effectively fill the first of 16 layers.
Notice how the balls basically comprise two interwoven arrays of 9 balls each. One set coloured Red, the other coloured Orange.
And notice the spatial dimensions - which apply in both the X and Y dimensions
The following slides will develop the subsequent layers and the related dimensions
The fourth slide (above) only shows the first ball in the second layer for the sake of clarity.
The dimensions in the X-Section at the bottom of the slide are very simple.
The lateral spacing of the red balls is 1.6
The diagonal spacing of the red/blue balls is 1.0
Hence by Pythagoras (3/4/5 triangle) the vertical spacing of the first and second layers is precisely 0.6 (0.6/0.8/1.0)