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Enigmas & puzzles

By Ian Stewart   December 2005

Monumental calculation

Klephtnose III’s propitiously proportioned pyramid was almost complete, with just a few casing stones missing, and the pharaoh was contemplating a new addition to the project. Once more he summoned his master of monuments, Amunaleg.

“It is said that your skill in numbers is unsurpassed in my kingdom.”

“My skills, such as they are, are always at your command, O mighty king.”

“A pyramid has eight edges, does it not?”

“That is so. Four around its base and four slanting to its tip, my lord.”

“It has amused me to imagine that a sacred number might be assigned to each edge, carved into the white limestone of the casing near its midpoint, in honour of the god Poli-Hed-Ra. Now, a pyramid has five corners, does it not Master Amunaleg?”

“Indeed it does, my lord. Four at the base and one at the apex.”

“I require the numbers assigned to those edges that meet at any corner to total the sacred 16,” said Klephtnose. “Might that be possible with consecutive numbers, 1 to 8?”

The architect flinched inwardly. “O mighty king, 5 times 16 is 80, and each edge meets two corners, so the total of the numbers on the edges must be 40. However, the sum of the numbers 1 to 8 is only 36, which is too few.”

The pharaoh smiled. “My own deduction, precisely,” he said. “So I will permit you to choose eight different whole numbers from 1 to 10. But choose wisely!”

What numbers did Amunaleg choose, and how did he number the edges?

Scroll down for the answer

The answer

There are three sets of eight numbers between 1 and 10 that add to 40:
1 2 3 4 5 7 8 10
1 2 3 4 6 7 8 9
1 2 3 4 5 6 9 10
By considering where the largest number 10 goes, and exploring possibilities, the first two sets can be eliminated. The
third set gives a unique answer (aside from rotations and reflections of the pyramid):

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