Imagine a cube with a positive integer written on each of its six faces. Now imagine cutting the vertices of this cube; the resulting solid is known as a truncated cube, or truncated hexahedron, and is shown in the image. The six original squares are now octagons, but the integers remain unchanged, and the eight vertices have become triangles. Let the number on each triangular face be the product of the three numbers written on the octagons that share an edge with it.

The sum of the numbers on all the triangles is equal to 6006. Find the smallest possible sum of the original six numbers written on the faces of the cube.

[This question has been adapted from 104 Number Theory Problems: From the Training of the USA IMO Team]

**You may discuss the question below, but no full answers, please, until the competition has closed!**

**You can receive these questions directly to your email box or read them in an RSS reader. Subscribe using the links on the right.**

**If you would like to DONATE PRIZES for the next PMQ, send me a message here.**

**How to Enter****Send your complete solution by email to pmq18feynman@giftedmaths.com.**This email address shall be removed after the competition closes to avoid spam.

**This PMQ18 competition closes on MONDAY 13 May at 23:59 GMT - one extra weekday from now on**.

**The Prize****The prizes for this PMQ are 3 free places in our Online Maths Club for ONE YEAR.**The very first correct solution will receive a prize plus two others randomly selected from all the other correct answers. The email time stamp shall determine the order of entries received. All winners can have their name posted and a link to their own online profile at their favourite social network or their own blog.

**Quick Rules****Look at the expanded rules on our PMQ page.**

Anybody can enter our Prize Maths Quiz; adults and students.

Use the official server clock in the right column to avoid late entries.

All emails and email addresses sent to us will be deleted after the winners have been processed.

DO NOT submit your entries to the comments section at the bottom of this post.

You CAN discuss it there but you must email us to enter the competition.

We adhere to COPPA guidelines regarding children's online safety and security.

Enjoy the challenge!

**Send us your solution! You can't win if you don't participate.**

Then tell your friends!