## POTW #14 - April 8, 2024

**MCR Problem of the Week, April 8, 2024 Edition**

In a distant land, there are three wise scholars: Althea, Balthazar, and Cyrus. Each scholar wears a hat, which is either blue or red. They know that at least one hat is blue and at least one hat is red. However, they cannot see the colour of their own hat, only the hats of the other two scholars.

Althea says, "I do not know the colour of my own hat."

Balthazar then says, "I do not know the colour of my own hat."

Finally, Cyrus says, "I know the colour of my own hat."

**The scholars cannot hear each other.**

Assume that all three scholars are truthful and logical. A "state" is defined as a three-digit binary number where `1`

represents a blue hat and `0`

represents a red hat. The first digit represents Althea's hat colour, the second digit represents Balthazar's hat colour, and the third digit represents Cyrus' hat colour.

Given the statements of the scholars, what is the sum of all valid states (in base 10)?

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