Brain Teasers are Back!

Hi IB,

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 Trains

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 :sunglasses:

Perhaps the diagram above will help/inspire a result. This one really is straight forward.

A small amount of Newton’s idea of relative velocities is all that is needed.

And let’s face it, we encounter the “overtaking” version on our motorways every day when two HGVs are running side-by-side !!

Let the speed of Train A = A ft/sec
Let the speed of Train B = B ft/sec

A + B = 120 ft/sec (1)
A – B = 40 ft/sec (2)

(1) + (2) gives 2A = 160 ft/sec
Hence A = 80 ft/sec

Sub for A in (2) gives B = 40 ft/sec

Converting these to vulgar fractions is not for the faint-hearted youngsters of today !

I’ll leave this easy one for another day.

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Well, it’s now another day and the answer to the balls-in-the-box teaser is 288.

Sorry Don, didn’t see your response till now.

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 did that before scrolling further and seeing your last post above. I’d be interested in seeing the solution!

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 :sunglasses:

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.

You are correct in that the bottom layer doesn’t fill the entire space.

But each successive layer likewise, doesn’t entirely fill the space for that layer.

It will probably take one of those “Eureka” moments of idle thought to see a solution.

I’m too busy dwelling on climate change now :joy::joy::joy:

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 :sunglasses:

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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 :sunglasses:

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 second slide (below) shows a single ball in the second layer. It sits evenly on two red balls and two orange balls.

You can probably already visualise where all the remaining 17 balls will fit in this second layer, but i’ll post that in the next slide.


The third slide (above) shows the second layer with 9 blue balls and 9 yellow ones.

You will notice that the first and second layers are essentially the same, with the same spacing.

Hopefully, you can visualize all 16 layers as alternating sets of the first and second layers ?

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)

In this 5th slide, I have turned the box onto its side (it was easier to draw !)

I have added the dimensions of the various layers which hopefully you can see, add up to 10.0 ie 2x0.5 ends plus 15x0.6 layers

QED

Hope you found it enjoyable.

And again, many thanks to Matthew T who posted this teaser 15-20 years ago.

Well, I think it was Matthew, so sincere apologies if my memory of names isn’t as good as my memory of the teasers !

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As before, each circle contains a number which is the sum of the two circles beneath.

Simply fill in the missing numbers !