Fred's maths problems
If the statement is not true, what can we say will be true?
Problem
Consider the statement about Fred:
(*) Every day next week, Fred will do at least one maths problem.
If statement (*) is not true, which of the following is certainly true?
(A) Every day next week, Fred will do more than one maths problem.
(B) Some day next week, Fred will do more than one maths problem.
(C) On no day next week will Fred do more than one maths problem.
(D) Every day next week, Fred will do no maths problems.
(E) Some day next week, Fred will do no maths problems.
(F) On no day next week will Fred do no maths problems.
This question comes from TMUA Specimen paper 2.
The Test of Mathematics for University Admission (or TMUA) is designed to give you the opportunity to demonstrate that you have the essential mathematical thinking and reasoning skills needed for a demanding undergraduate Mathematics or Mathematics-related course. There are several UK universities which encourage their applicants to sit TMUA, and a good performance may result in a reduced offer.
For more information about TMUA visit https://esat-tmua.ac.uk/about-the-tests/tmua-test/.
Student Solutions
Marcus from Westminster School in the UK sent in this explanation:
If you imagine that it's true, and you think about what would be required to make it false, you would see that making him do more math tests on one day would not work, as it is AT LEAST. [Therefore], you must do none.
Michael from German Swiss International School in Hong Kong sent in a more detailed explanation:
The statement (*) is only true when Fred does at least one maths problem every day next week. Therefore, the statement (*) is false when Fred does no maths problems on at least one day next week.
First consider F. F states that on no day next week will Fred do no maths problems. This is the equivalent of "Fred will do at least one maths problem every day next week." Therefore this is equivalent to the statement (*).
Now consider A. A states that Fred will do more than one maths problem every day next week. If the statement (*) is true, A is true. In fact, the converse of this is true: If A is true, then (*) is true. Therefore if (*) is not true, A cannot be true either.
Then consider C. C states that on no day next week will Fred do more than one maths problem. Even if the statement (*) is false, C doesn't always hold true. This is because Fred could do no maths problems on some days, and more than one maths problem on other days.
To rule out B from being the correct answer, we could assume that Fred does no maths problems every day next week.
D is also not the correct answer. Even if Fred doesn't do at least one maths problem every day next week, he could still have done some maths problems on some days.
With all of the other choices eliminated, E is the only correct answer. The only way for Fred to not do at least one maths problem every day, as stated in the statement (*), is for Fred to do no maths problems on some days next
week.
The Test of Mathematics for University Admission (or TMUA) is designed to give you the opportunity to demonstrate that you have the essential mathematical thinking and reasoning skills needed for a demanding undergraduate Mathematics or Mathematics-related course. There are several UK universities which encourage their applicants to sit TMUA, and a good performance may result in a reduced offer.
For more information about TMUA visit www.admissionstesting.org.
Teachers' Resources
This problem and the others in KS5 Proof shorts are great as lesson starters, warm downs or for stimulating discussion during a lesson. This problem helps students to understand how to construct the precise negation of a statement. This is the first step in proof by contradiction, and exploring the concept in the simple context of this problem will help students to clarify their thinking.