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# Two objects, Three views, Identical shapes

### #1

Posted 17 December 2013 - 07:38 PM

Object A appears in each case to be a square of side a.

Object B appears in each case to be a circle of diameter a.

If they are made of the same material, can you say which object is heavier?

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #2

Posted 18 December 2013 - 12:34 AM

### #3

Posted 18 December 2013 - 12:03 PM

Spoiler for

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #4

Posted 18 December 2013 - 04:56 PM Best Answer

**Edited by harey, 18 December 2013 - 04:58 PM.**

### #5

Posted 19 December 2013 - 05:22 PM

Going a bit further just for fun, which object can weigh the most, and which can weight the least?

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #6

Posted 20 December 2013 - 12:40 PM

The circle is inscribed to the square...

Going a bit further just for fun, which object can weigh the most, and which can weight the least?

That's easy. The circle is inscribed to the square.

### #7

Posted 20 December 2013 - 02:55 PM

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #8

Posted 20 December 2013 - 08:54 PM

The best way is to make them of equal volume ( suppose what I call a "3D square" is a prism?):

v_prism=a*a*d_prism

v_cylinder=pi*a/2*a/2*d_cylinder

=> d_cylinder=(4/pi)*d_prism

While it might work for small d, with growing d, it will became more and more perceptible that the cylinder should be a kind of barrel (or a pancake). The formulae get too complicated for my taste and my possibilities.

### #9

Posted 20 December 2013 - 10:44 PM

^{2}d.

The lightest "sphere" could be a "soap bubble'" Volume = pi a

^{2}d.

In that sense, the "cube" could be the lighter of the two.

Just a thought.

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #10

Posted 21 December 2013 - 06:03 PM

Yes, I see. There must be a lot of solutions, like taking a sphere and a cube and drilling holes into the cube until it becames lighter than the sphere - no formula, no misscalculation.

My preferred solution is still that of the three squares and three pankaces in the planes xy, xz and yz.

BTW, the Volume=3a^{2}d**-d ^{3}** - I have the same oversight in my post n. 8.

I just wonder how often we make false deductions in the real life in the style "three projections=circle => it must be a sphere".

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