The twins were walking on a trail near Dog Creek when they first met him. The local police had just told the twins that they were looking for a major drug shipment, possibly arriving on a numbered bus.
"Something strange about the number on a bus I saw coming through here," said Craggy without introduction and quite coincidentally. "This bus number has two digits and the sum of those digits equals the sum of the bus number's prime factors."
Problem 1. Cloe and Eli quickly figured out what the number on the bus must be. Can you?
Hint: Consider the number 45. The prime factors are 3, 3 and 5. So the sum of the prime factors is 3 + 3 + 5 = 11. The sum of the digits is 4 + 5 = 9. So, 45 is not the answer.
The twins reported the bus number to the police, who found the bus and traces of drugs. The police were convinced, however, there were other buses involved in the scheme. The twins went looking and found Craggy perched atop the volcanic plug called French's Dome. "Yes, I found another strange one," he said as he awoke from his reverie. "Also two digits. The sum of the digits is half the sum of its prime factors."
"That's not enough information," Cloe protested.
"Right," said Craggy. "The bus number in question also has only two prime factors."
"I've got it," said Eli.
Problem 2. Which bus number did Eli calculate?
"You are very clever," Craggy said with his rough smoker's chuckle. "There's another two-digit bus number where the sum of the digits is greater than the sum of its prime factors and the number of prime factors is not 5."
"That's not enough information either," said Eli.
"Ok, I'll nail it down by telling you how many prime factors it has," said Craggy.
"No need," Cloe interrupted. "I know what it is."
Problem 3. What number did Cloe say? How did she know?
1. If the sum of the digits equals the sum of the prime factors, the bus number can only be 27. The prime factors are 3, 3 and 3. 2 + 7 = 9 and 3 + 3 + 3 = 9. Clearly, large prime factors such a 23 can't play a role, because two digits can add up to a maximum of only 18 (= 9 + 9). If your child is older, you can help him or her write a program to generate the factors. Younger children should be told "find the factorization of the numbers between 20 and 30" as an extra hint.
2. When the sum of the two prime factors is twice the sum of the digits, the bus number can be only 91. The prime factors are 7 and 13. That's the only number meeting the criteria having only two prime factors. Younger puzzlists may want to know that the sum of the primes is 20.
3. The two-digit numbers whose digit sum is greater than the prime factor sum are 18 (factors 2, 3 and 3), 48 (2, 2, 2, 2 and 3), 96 (2, 2, 2, 2, 2 and 3), 98 (2, 7 and 7) and 99 (3, 3 and 11). Craggy said the number of prime factors was not five, so 48 is out. Because Craggy was "nailing it down" by telling Eli how many prime factors the number had, it must have a number of factors different from the others on this list. All of the numbers except 96 have only three prime factors, so Cloe inferred that 96 was the answer.
Note to parents: The idea of this puzzle is to get children to play with prime numbers. You might start by helping them factor some numbers. Then let them find the answers to the mysteries, perhaps with the extra hints.