You are at a party and overhear a conversation between Lucy and her friend.
In the conversation, Lucy mentions she has a secret number that is less than 100.
She also confesses the following information:
"The number is uniquely describable by the answers to the following four questions:"
Q1) Is the number divisible by two?
Q2) Is the number divisible by three?
Q3) Is the number divisible by five?
Q4) Is the number divisible by seven?
She then proceeds to whisper the answers to these questions to her friend. Unfortunately, because of the ambient noise at the party, you only hear the answer to one of the questions. However, knowing just this one answer allows you to determine Lucy’s secret number.
Questions
Which question and answer did you overhear?
If the answer to this question is “Yes”, what is Lucy’s secret number?
Reveal Answers
Solution
Since there are four questions, and each answer can be yes or no, there are sixteen possible combinations of answers (Think of these like binary bits of a four bit number).
Here are are all sixteen of these answer combinations enumerated and, on the right, which 'secrets' would trigger these answers.
| Divisible | 7 | 5 | 3 | 2 | Solutions |
|---|---|---|---|---|
| N | N | N | N | 1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 |
| N | N | N | Y | 2, 4, 8, 16, 22, 26, 32, 34, 38, 44, 46, 52, 58, 62, 64, 68, 74, 76, 82, 86, 88, 92, 94 |
| N | N | Y | N | 3, 9, 27, 33, 39, 51, 57, 69, 81, 87, 93, 99 |
| N | N | Y | Y | 6, 12, 18, 24, 36, 48, 54, 66, 72, 78, 96 |
| N | Y | N | N | 5, 25, 55, 65, 85, 95 |
| N | Y | N | Y | 10, 20, 40, 50, 80, 100 |
| N | Y | Y | N | 15, 45, 75 |
| N | Y | Y | Y | 30, 60, 90 |
| Y | N | N | N | 7, 49, 77, 91 |
| Y | N | N | Y | 14, 28, 56, 98 |
| Y | N | Y | N | 21, 63 |
| Y | N | Y | Y | 42, 84 |
| Y | Y | N | N | 35 |
| Y | Y | N | Y | 70 |
Remember that Lucy said that the answers to the questions uniquely determine her number? This means we are looking for rows that have just one solution. As it happens, there are only two; the last two rows. The secret is therefore 35, or 70.
If you look closely, you will see that both of these solutions have the same answers to the questions "Divisible by seven?", "Divisible by five?" and "Divisible by three?". Because of this, we don't need to know the answers to these questions!
The question we need to know the answer to is "Divisible by two?"
As we heard this answer is "Yes", then Lucy's secret number is 70.
