 Try this beautiful problem from AMC 10A, 2007 based on Numbers on cube.

## Numbers on cube – AMC-10A, 2007- Problem 11

The numbers from $1$ to $8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?

• $16$
• $18$
• $20$

### Key Concepts

Number system

Cube

But try the problem first…

Answer: $18$

Source

AMC-10A (2007) Problem 11

Pre College Mathematics

## Try with Hints

First hint

Given condition is “The numbers from $1$ to $8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same”.so we may say that if we think there is a number on the vertex then it will be counted in different faces also.

can you finish the problem……..

Second Hint

Therefore we have to count the numbers $3$ times so the total sum will be $3(1+2+….+8)$=$108$

can you finish the problem……..

Final Step

Now there are $6$ faces in a Cube…..so the common sum will be $\frac{108}{6}$=$18$