Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 Which kind of claim is knocked down by a single counter-example?
Show worked solution
A "must" claim asserts every case, so one exception is fatal. A "could" claim only needs one case to work, so a counter-example does not touch it.
2 Two whole numbers add to an odd total. Write the number of them that must be odd, and the number that must be even.
Show worked solution
Odd + odd and even + even both give an even total, so an odd total needs exactly 1 odd and 1 even.
3 Give a number that is divisible by 4 but not by 8, using the smallest one you can.
Show worked solution
4 itself: it is divisible by 4, and 4 ÷ 8 is not a whole number.
Level 2 · Core
4 "If a shape is a square, then it has four sides." A shape has four sides. What follows?
Show worked solution
Rectangles and other quadrilaterals also have four sides, so it could be a square, but need not be. Concluding it must be one is reversing the statement.
5 Ana, Ben and Cal each have one of red, blue and green. Ana is not red and not green. Cal is not red. Write Ana's colour and Ben's colour.
Show worked solution
Ana must be blue. Red and green remain for Ben and Cal, and Cal cannot be red, so Ben is red and Cal is green.
6 Three cards are turned over from a pile and their sum is even. How many of those three cards must be odd, if it is not zero?
Show worked solution
Three odds give an odd total and one odd gives an odd total, so it must be 2 odd and 1 even (or none odd at all).
Level 3 · Challenge
7 Five cards lie face down: three odd and two even. Three are turned over and their sum is even.1 Exactly two of the three turned over must be odd.
2 All three turned over could be even.
3 The two left behind must include exactly one odd card.
Which statements are correct?
2 All three turned over could be even.
3 The two left behind must include exactly one odd card.
Which statements are correct?
Show worked solution
An even sum from three numbers needs 0 or 2 odds, and with only two evens available "no odds" is impossible, so exactly 2 are odd (1 ✓) and all three cannot be even (2 ✗). That leaves 1 odd and 1 even behind (3 ✓). 1 and 3.
8 Consider "if it rained, the match was cancelled". Write YES or NO for each: does "the match was not cancelled" tell you it did not rain, and does "the match was cancelled" tell you it rained?
Show worked solution
Yes — if it had rained the match would have been cancelled, so no cancellation means no rain. No — something else could have cancelled it. Reversing an "if … then" is not allowed.
9 A whole number is divisible by 6. What is the smallest such number that is NOT divisible by 4?
Show worked solution
6 — it is divisible by 6, and 6 ÷ 4 is not a whole number.