Tom's number
Problem
"Guess my number," said Tom, "I'll only answer yes or no."
"OK. Is it less than $1000$?" "No."
"Is it less than $2000$?" "Yes."
"Is it even?" "No."
"So it's an odd number between $1000$ and $2000$?"
"Yes, but you knew that already. It's a wasted question!"
"Is it divisible by $3$?" "No."
"Is it divisible by $5$?" "No."
"Is it divisible by $7$?" "Yes."
"If I divide it by $7$ do I get a number that is less than $100$?" "No."
"Do I get a number that is less than $200$?" "Yes."
"Ah! $100 \times 7$ = $700$ and $200 \times 7 = 1400$. So your number is less than $1400$?"
"Yes, but that's another wasted question."
"Ha! $14$ squared is $196$.
So your number is $7$ times the product of two prime numbers less than $14$?"
"Yes."
"I am sure I know your number and this question is just checking.
Is your number palendromic? Does it read the same in both directions?"
"Yes."
What was Tom's number?
Getting Started
You could start at either end of the question.
Try making a list and eliminating numbers that do not fit.
Student Solutions
We knew a great deal about Tom's number, and people made good use of the information to figure out what the number was exactly.
Kevin from Etobicoke, Ontario, Canada narrowed down the answer by following the clues step by step. Once he was sure he had Tom's number, he returned to the clues to prove that his answer was the correct one. Thomas from Tattingstone Primary, near Ipswich, thought of possible answers and then arrived at his final answer by eliminating the numbers that would not work for the clues given. Holly and Joanne, from Moorfield Juniors, used a technique called guess and check; they chose a number that they estimated was reasonable and then checked it against each of the clues.
Good detective work led many of you to finding out what Tom's Number was. Ben, Malcolm and Paul from Yarm Primary School, Jake and Ben of Moorgate Primary in Tamworth, Staffordshire, Zoe of Eastbury Farm School, and classmates Matt, Adam, Dave and Chris all found the answer.
The solutions sounded very much like entries from Sherlock Holmes. Anis wrote:
From the information that was stated I can conclude that the number is:
- Odd
- Divisible by 7
- Between 1000 and 1400
- 7 times the product of two prime numbers less than 14
- Palendromic
Once I knew the number was palindromic I tried dividing 1331, 1221, 1111 and 1001 by 7. 1001 was the only one that gave a whole number of 143. I divided that by all the odd numbers between 7 and 11. 11 gave me the answer of 13, and as both are prime numbers, I knew that 1001 was correct.
Molly from Oaklands Primary School, Biggin Hill agrees with the solution. She sent this pdf of her reasoning. Well done Molly!