Brain Teasers ? or 50 Years On........... ?

OK, here's my working. Different method to Don, but the same answer.

We can't both be wrong, can we ?

Hi Don, I think your method seems valid, but why not just impose the alpha = beta condition right away and get the answer directly ? 

Ian 

Ah, Ian. I was uncertain about the geometric movement. I wasn't certain that the "pivot" points remained fixed or that the cones, when being rotated remained "jambed free". So I tried a few specific rotation angles to check.

Before long, I realised that somewhere around the 27 degree mark, it would be relatively easy to estimate when Alpha = Beta.. All I then had to do was iterate a couple of times to refine the accuracy.

But, I might yet try setting "Alpha" = "Beta" and do it "properly" !!

Then again..............

The equality that I used when tracking the value of β, for varying values of α was :-

β = Atan [Cos α ÷ √3]

Time for a bit of light relief...........

How many of the following statements are true ?

None of these statements are true

Exactly one of these statements is true

Exactly two of these statements are true

All of these statements are true

A0  B1  C2  D3  E4

B1

The first and last statements are mutually exclusive (statement 1 is self contradictory)

statement 3 can't be true as it would render statement 2 false. Therefore only one statement can be true.

      Alpha          Alpha          Alpha          Beta          Beta          Apex          β - α

       Deg          Rads                       Cos          Rads          Deg            Angle          Rads

Abstract from my calculations. Hopefully this time the headings and columns are aligned and the numbers are still readable.

Alpha = Beta with an Apex Angle somewhere between 54.36811° and 54.36769°.

54.368° looked pretty close to me !

sjbabbey posted:

B1

The first and last statements are mutually exclusive (statement 1 is self contradictory)

statement 3 can't be true as it would render statement 2 false. Therefore only one statement can be true.

Nice explanation sj, and bang on with the answer. Well done.

Sorry for the silence but have been bogged down in work.

I was trying first trying to find the solution and was drifting into planes and angles between them and a whole host of complicated messy equations. In an effort to simplify the solution I did get to them same answer as above (54.368 degrees) but I don't believe it is correct. In my simple workings I believe there is an assumption for any one cone that the line of contact with thetwo other cones either side defines a plane that passes through the centre of the cone. This is an approximation that works for small angles but as the apex angle gets bigger the the plane defined by the lines of contact will move away from the centre of the cone.

Still trying to get to the real answer....

Desk is covered in scribbled on paper.

Right, think the answer is 53.13 degrees [2 x arctan 0.5].

Getting towards an elegant proof...

Involves finding the equations of planes that 'box' the cone in and calcualting the angle between them.

Need to do some proper work now! Solution will come later.

Matthew T posted:

Sorry for the silence but have been bogged down in work.

I was first trying to find the solution and was drifting into planes and angles between them and a whole host of complicated messy equations. In an effort to simplify the solutionI did get to the same answer as above (54.368 degrees) but I don't believe it is correct.  In my simple workings I believe there is an assumption  for any one cone that the line of contact with the two other cones either side defines a plane that passes through the centre of the cone. This is an approximation that works for small angles but as the apex angle getsbigger  the plane defined by the lines of contact will move away from the centre of the cone.

Still trying to get to the real answer....

Desk is covered in scribbled on paper.

 

You share my own doubts about 54.368.

I believe you are correct about the lines of contact between three cones. They don't define a plane passing through the centre of the middle cone, except in the case the cones' altitudes all lie in the same plane, eg horizontal. They represent the tangent points between the centre of the middle cone and the two contact lines with the cones either side. As the six cones become more and more vertical, this becomes more obvious.

I believe the Apex Angle gets progressively smaller as the cones translate from a horizontal group (apex angle = 60 deg, similar to the spokes of a wheel) towards a vertical group (apex angle = zero, similar to a ring of roadside cones).

I visualize :-

the initial state as six cones with their Altitudes all lying horizontal. The Altitudes lie in a common, horizontal plane. The upper slopes lie around the surface of a conical plane and likewise, so do the lower surfaces.

All six cones then rotate downwards. In doing so, the Altitudes begin to lie on the surface of a conical plane, as do both the upper and lower slopes.

At a certain point, the conical plane containing the lower slopes becomes horizontal (or flat). This is the stage that Matthew 's puzzle wants us to define !

Thereafter, the upper slopes, the Altitudes and the lower slopes each lie around the surface of three "concentric cones",

until first, the lower surfaces lie around the surface of a vertical cylinder,

and shortly thereafter, the Altitudes likewise lie around the surface of a vertical cylinder, diameter 2R where R = radius of the base of each of the six cones.

Don't know if this helps, or even if my visualization is representative of reality !!

Frank, how did you arrive at 53.54 ?

Eliminating my 57.3 (Radian) and Ian's self-declared mistake at 52.7 I have calculated an "average"...

Frank 53.54; Ian 50.42, 54.37: Don 54.37; Matthew 53.13

Average = 53.16

I'm sure the Pure Maths Dept at Cambridge will not be impressed with this latest approach.............

Puzzles............

If you have successfully tackled one or two of the Brain Teasers or maths questions in this thread, (even if you haven't posted), I need hardly explain why such puzzles can be so enjoyable.

They reel you in. Distractions around you disappear as you concentrate on finding a solution. Being forced to use your wits or conjure up distant recollections is life-affirming. And deductive reasoning in simple, logical steps is reassuring, especially when real life is often so illogical. Good puzzles present achievable goals, which are supremely satisfying when they are resolved.

On the other hand, they can be bloody frustrating !!!

I agree there is the 'assumption' of where the  line of contact lies but I'm not convinced it is wrong ( yet ?). 

Surely no matter what the apex angle of the cone,  as seen from above, the cones must touch at the extremities (in the plane of the table). Surely this forces the plane defined by the touching lines and the apex to intersect the middle of the cone ? 

Another way of saying the same thing is that the distance between the points of touching at the base of the cone is the diameter of the base so by definition must run through the centre of the cone. 

We need a good diagram ! 

Ian 

Ian G. posted:

I agree there is the 'assumption' of where the  line of contact lies but I'm not convinced it is wrong ( yet ?). 

Surely no matter what the apex angle of the cone,  as seen from above, the cones must touch at the extremities (in the plane of the table). Surely this forces the plane defined by the touching lines and the apex to intersect the middle of the cone ? 

Another way of saying the same thing is that the distance between the points of touching at the base of the cone is the diameter of the base so by definition must run through the centre of the cone. 

We need a good diagram ! 

Ian 

 

Ah interesting, I'm wrong again , as 5 mins playing with two coins have persuaded me - Thanks Matthew T!

Matthew T posted:

Right, think the answer is 53.13 degrees [2 x arctan 0.5].

Getting towards an elegant proof...

Involves finding the equations of planes that 'box' the cone in and calcualting the angle between them.

Need to do some proper work now! Solution will come later.

"elegant" !!!!!!!!!!!

I think that if Ian or myself could come up with any sort of proof we'd be happy

....well, I would be ! (I'll let Ian speak for himself)

I think I see where you are coming from Matthew.

There are three vertical intersecting planes (60 degrees apart) that meet in a vertical line at the Apex of the six cones.

A fourth, horizontal plane contains the Altitude and a base diameter of each of the six cones when they are lying horizontal.

Rotate the cones downwards until the lower sloping side of each cone lies on a horizontal table which defines a fifth plane, parallel to the fourth plane.

Latest thoughts..........

The contact line on the sloping side of a cone, where the cone touches the adjacent plane, changes as each cone is rotated downwards from the horizontal.

The two contact lines on each cone are initially from the end of a diameter of the base, to the Apex,  (when the cone is horizontal). And their distance from the intersection of the planes is 2R (ie diameter of a base)

However, when a cone is vertical, the two contact points around the base of a cone, have moved to a point 30 degrees towards the intersection line of the three vertical planes. The intersecting vertical planes are tangential to the base of each cone, at points which subtend an angle of 120 degrees around the base of the cone.

The points on the vertical planes at which the base of the cones now touch when vertical, are probably R√3 from the intersection of the planes.

Back to the drawing board !!!!

Don Atkinson posted:

Frank, how did you arrive at 53.54 ?

Don Atkinson posted:

Frank, how did you arrive at 53.54 ?

Not very accurately. I made the mistake of not making a sketch before I made the calculation.

 I was working on the idea below.

There is a line of contact between a pair of cones. The line of contact is, from the cone point (A) to the base of a cone (B). When you look down on the cones, the line of contact appears to be the same length as the cone base, but it’s longer due to the fact it’s at an angle.

After many screwed up pieces of paper, I’ve not come up with a formula, but I’m pretty sure 54.37 is correct.

Although, looking at the bottom diagram, I’m not sure it’s correct, it won't actually look like that when viewed on plane x-x

Thanks Frank. Difficult problem to visualize, never mind solve !

I have my doubts about 54.37, (and always have had doubts)

Ian suggested working with tuppenny pieces and having some sort of diagram........

Good idea Ian !

In the above sketches I have only drawn the BASE of the cones (tuppenny pieces !)

The ones on the right illustrate the cones when their Altitudes are horizontal. The ones on the left, with Altitudes vertical.

Matthew is looking for the situation where the Apex (omitted for clarity !)  is APPROX 27 degrees down, ie in between my two extremes, with the SLOPING side of the cone horizontal.

I don't know if this picture helps, I think it helps me realize the problem is a bit more complex than I previously thought................

PS If any of my dimensions are wrong, please shout !!!

As a break from looking at cones, here is a genuine practical problem from a party last night.

• The host had invited 16 guests, and the host had prepared a team quiz which had 4 rounds.
• The host wanted 4 teams of 4 people for each round.
• For each successive round, teams were to be reorganised so that each member had new team-mates - ie their new team had no member with whom they had played previously.
• The objective was to ensure that, by the fourth round, everyone had "played" with everyone else.
• All went well for the first 3 rounds, but for the fourth round we couldn't get a solution which avoided a number of "repeat members".
• The host looked at me, as the supposed maths expert, to confirm that it should have been possible to achieve her objective. I was unable to come up with a solution which worked, but neither was I able to explain why it was impossible.

Any suggestions gratefully received!

Steve D