Clued In

Ageless Advice

**1. [Easy]**Two dads meet at work. "If you multiply the ages of my three children, you get 35," says one dad to the other. "No two are the same age. How old are they?"

**2. [Easy]**Two moms meet at work. "If you multiply the ages of my three children, you get 36," says one mom to the other. "No two are the same age. How old are they?"

"I don't know. That's not enough information," says the other mom.

"OK, here's a clue," says the first mom. "The sum of their ages is 11."

**3. [Difficult]**A mom and a dad meet at work. "If you multiply the ages of my three children, you get 36. How old are they?" asks the mom.

"I don't know. That's not enough information," says the dad.

"The sum of their ages is the number on that sign."

"That's still not enough information."

"Only the youngest has red hair."

"If that's true, I know how old your children are."

Can you figure out the ages of the three children based on this conversation? Hint: We don't know what the number on the sign was, but we do know that knowing the sum is not enough to determine the children's ages.

**4. [Difficult]**In the previous problem, suppose multiplying the ages of the three children yielded 72 instead of 36? What would the three ages be?

**5. [Very difficult]**What is the next number after 72 that could replace the 36 in this puzzle and still give the dad enough information to know the children's ages?

Tomb Etchings The notebooks of archaeologist Dusty deBrie contain the four intriguing sketches above. Each sketch constitutes part of a larger design that had been uncovered on one wall of the lost tomb of King Foraday III. DeBrie's notes state that every 3-by-3 sketch can be found within a 5-by-5 grid of black and white squares. And every square in the 5-by-5 grid appears in at least one of the sketches. Only the first sketch is definitely in the same orientation as it appears in the larger grid. Sketches 2 through 4 may be rotated.

**1. [Easy]**Can you reconstruct the original 5-by-5 pattern?

**2. [Challenging]**DeBrie's rival, Doug Allot, argued that the original pattern was not 5 by 5 but five units wide by four high. He agreed, however, with everything else. Can you reconstruct the original if Allot is right? (There is more than one answer.)

**3. [Challenging]**Scholars at the Institute for Buried Treasure believe that the original pattern was not 5 by 4 or 5 by 5 but 4 by 4. They, too, agree with DeBrie's other conjectures. Can you reconstruct the 4-by-4 pattern?

Do you know your ABCs?

**1. [Easy]**There are nine letters in the 3-by-3 grid below. Which of the following six rules does this arrangement of letters obey? (Letters that are situated diagonally from each other are not considered adjacent.)

a. Adjacent letters are always different colors.

b. A is never adjacent to B.

c. C is never adjacent to a yellow letter.

d. The top left letter is a B.

e. The top middle letter is red.

f. Two of the A's are adjacent to the third A.

**2. [Difficult]**Rearrange the letters in the grid so they obey all six rules. There is only one solution. Hint: Start with rule c.

**3. [Challenging]**Which of the rules is redundant? In other words, which rule could you remove and still have only one possible solution? There may be more than one answer.

**4. [Challenging]**Which of the rules could you remove and still have only two possible solutions? There may be more than one answer.

**5. [Difficult]**By itself, rule d reduces the number of legal arrangements to one-third of the total number possible, since only one out of three of all the possible arrangements obeys this rule. Which rule by itself reduces the number of possible answers to the smallest number?

**Solution**

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