perplexus.info :: Just Math : Balanced Stellar

Balanced Stellar (Posted on 2007-06-22)

The graphic represents the partially exploded net of a stellated cube as viewed in its cubic layout; each face is a right pyramid viewed from above.

As with a problem of similar concept the letters are placeholders for numerals; in this case they are 1 through 24, [A=1].

The apices of the pyramids form 'external' vertices
while those which describe the foundation cube are the 'interior' vertices.

1. What is the global sum of the 6 'external' vertices and then their average?
2. What is the global sum of the 8 'interior' vertices and then their average?
3. How can the faces be labeled to optimise both #1 and #2 simultaneously?

The optimum outcome would be for all vertices to have the value of their type, 'external' and 'internal'.

Submitted by brianjn

Rating: 4.0000 (2 votes)

Solution: (Hide)

1. The 6 'external' vertices total 300 and thus the average is 50.
2. The 8 'internal' vertices total 600 and thus average 75.
3. Both criteria can be met in the layout. I offer a solution that fully meets those. Below it is one which goes close (4 numbers occupy slightly different positions 5 & 6 and 14 & 15 are exchanged).

My solution: A B C D E F G H I J K L 1 22 3 24 8 18 7 17 10 15 11 14 M N O P Q R S T U V W X 9 13 12 16 5 20 6 19 23 4 21 2
These layout as follows where the internal vertices are all 75:
_____ | 22 | |1 3| | 24 | | 18 | 15 | 13 | 20 | |8 7|10 11|9 12|5 6| | 17 | 14 | 16 | 19 | | 4 | |23 21| | 2 |

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Answer	K Sengupta	2024-01-18 03:04:49
Share time	brianjn	2007-11-05 01:29:25
Not quite perfect.	Leming	2007-06-29 10:43:35
re: 2 1/2 out of 3.	brianjn	2007-06-22 20:06:37
re: Almost there, almost there	brianjn	2007-06-22 20:02:44
1/2 separately	Charlie	2007-06-22 12:30:30
Almost there, almost there	Leming	2007-06-22 11:59:30
2 1/2 out of 3.	Charlie	2007-06-22 10:09:27

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