10*10*10 one ohm 3D grid.

1.3227*Unit('Omega')

The answer of 6 points. r=32/sqrt(15) at the date 2022/03/29.

One of the answer by Maple2022.

@Tokoro Number of resistors used to make one-ohm is > 20.

@dharr 

Parallel 2 one-ohm + Parallel 4 one-ohm + Parallel 8 one-ohm....

= 1/2 ohm + 1/4 ohm +1/8 ohm +...

= 1 ohm.

But each parallel resistors has same current 1 / number of parallel resistors.

Tokoro.  

@Carl Love Your post is true. The answer is two node resistor network and all resistors used have to be connected both ends and each current flows different ampritudes. The number of required one-ohm resistor is not infinite.

This is Electric circuit problem.

Tokoro. 

@dharr Smaller number of resistors is better answer.

We need more 1 ohm resistors to make one ohm. Only 4 resistores, we can't make one-ohm while each resistor must flow different current..

 "by using one-ohm resistors" means  @Joe Riel's answew is not included in the answer.

@mmcdara Second puzzle is find the minimum distance to measure. You can only move 6 surfaces of the box. Inside of the box, you can't move.

@dharr Thanks this is what I want to know how to use Maple.

By the way, unit cube is not the answer of 8 points for my 3D puzzle.

It has sqrt(2) and sqrt(3).

Tokoro.

@acer I want to know how to solve this puzzle by using Maple. 
If you can solve please show me the answer.

I have no idea how to solve this puzzle. 3D integer is so difficult for me. I can only check the distance is integer or not.

Tokoro.

@vv I think the one of inner distance is not integer. So, two times of your answer and 4,6,4,6,4 with radius 16/sqrt(15) is the answer I got.

Thanks a lot. You are special one to solve this puzzle. Next is 6 points on the circle.  

@Rouben Rostamian  Your answers are as same as my calculation results.

I am beginner of Maple. And to study these questions, I used Mathcad to solve. I found two cones have 0.691 rad and the third has the rest, the three cones have maximum volume point of 0.424. My interest is how to find these problems by Maple. Maple have so many functions and chance to serect the strategy to solve, therefore, I want to study how to use Maple for these problems.

My integer distance puzzle of 2D and 3D are very difficult to solve. Maple have many integer numbers functions. So, I ask the question. I hope some MaplePrimes users are interested to solve the Puzzles.

Tokoro.

@vv For me, this 2D puzzle is very hard and I can not find any answer from WWW or text books.

I found from n=2 to 100 the solutions of points on the circle. But has no idea to check the answers are smallest or not.

After find 2D problems, I make 3D version of this puzzle to solve. If you know or can solve this puzzle please show me the answer. 3D puzzle is so hard for me, but I have from 5 points  to 10 points answer for all integer distances on the sphere surface.

Tokoro.

@Carl Love Sorry, my explanation for the question is not good. vv shows the 4 points answer. Please try 5 points question.

Tokoro.

@Carl Love The unit square has sqrt(2) distance for diagonal points. sqrt(2) is not integer.

Tokoro.