Guess My Number: Clues That Do Real Work
Priya's table is playing Guess My Number. Priya writes four clues on the board: (1) My number is greater than 30. (2) My number is less than 50. (3) The two digits add up to 9. (4) My number is odd. Leo raises his hand first and says, 'It is 36, because 3 plus 6 equals 9.' Priya says, 'You used two of my clues, not all four.' The class is told to find every number that fits clues 1, 2 and 3 before anyone is allowed to answer. Then Mr. Adeyemi, the teacher, asks two harder questions. First: could Priya drop one clue and still have a game with exactly one answer? Second: Priya wants a different game whose answer is 36 instead of the current answer. What one new clue would do that without simply announcing the number? The class has ten minutes and must show its work for each question.
1. Find every number that fits clues 1, 2 and 3. Then apply clue 4 and name the answer. Show your list.
Read the worked answer
Clues 1 and 2 put the number between 30 and 50, so the tens digit is 3 or 4.
Clue 3 says the digits add to 9:
- tens digit 3: the ones digit must be 6, giving 36
- tens digit 4: the ones digit must be 5, giving 45
No other whole number works. 27 and 54 are outside the range, and the thirties and forties offer only those digit pairs that sum to 9.
So clues 1 to 3 leave exactly two candidates: 36 and 45.
Clue 4 says the number is odd. 36 is even, so it fails. 45 ends in 5, so it is odd. The answer is 45.
Leo's mistake was stopping at the first number that fit some of the clues, instead of listing every candidate and testing each one against all four clues.
2. Could Priya drop one clue and still have a game with exactly one answer? Test each clue by removing it and describe what happens.
Read the worked answer
No single clue can be dropped.
- Remove clue 1, greater than 30: 9, 27 and 45 all stay under 50, have digits summing to 9, and are odd. Three answers.
- Remove clue 2, less than 50: 45, 63, 81 and 99 all fit. Four answers.
- Remove clue 3, digits add to 9: every odd number from 31 to 49 fits. Nine answers.
- Remove clue 4, odd: 36 and 45 both fit. Two answers.
Each clue is load-bearing. The two range clues cut the possibilities down to 30 to 49. The digit-sum clue cuts that to two candidates. The odd clue picks one. Take any single clue away and the game either has many answers or exactly two.
3. Write one new clue that, added to clues 1, 2 and 3, makes 36 the only answer. The clue may not say 'the answer is 36' or name the number. Then explain why your clue works.
Read the worked answer
Any clue that is true of 36 and false of 45 will work. Three clean options:
- 'My number is even.' 36 is even; 45 is odd.
- 'The tens digit is 3.' 36 has 3 tens; 45 has 4.
- 'The ones digit is greater than 5.' 6 is greater than 5; the ones digit of 45 is 5.
Each of these separates 36 from 45. Since 36 and 45 are the only numbers that fit clues 1, 2 and 3, separating them leaves exactly one answer.
A clue such as 'My number is not 45' would also work, but it gives the answer away. That is why the task asks for a clue that describes a feature of the number instead of naming or excluding numbers.