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1. The four-digit number 27_4 is divisible by 3. Which digit could go in the gap?[4]
2. The two digits of a number are swapped and the number gets bigger by 36. Which of these could be the number?[4]
3. Which of these numbers can be divided exactly by both 4 and 9?[4]
4. A cube 3 units on each side is built from unit cubes and painted all over the outside. How many of the small cubes have no paint on them at all?[4]
5. Ana says: "Bo and I are both liars." Each child is either always truthful or always a liar. Who is telling the truth?[4]
A Ana onlyB Bo onlyC both of themD neither of themE it cannot be decidedKangaroo Atlas · https://kangaroo-atlas.vercel.app · seed 1 · CC BY-NC-SA 4.0 · Independent project, not affiliated with Math Kangaroo in USA, NFP.
1. B — 2 KA-0011 Divisibility rules: 2, 3, 4, 5, 9, 10
A I added the known digits to get 13 and then checked whether the missing digit itself was a multiple of 3.C I chose 3 because the question mentioned dividing by 3, without testing the digit sum at all.D I tested the last two digits, which is the rule for 4, not the rule for 3.E I added the digits but left out the 4 at the end, so my sum was four too small.2. B — 15 KA-0056 Place value and digit positions
A I checked that the digits differ and stopped, without working out how much the swap actually changes the number.C I saw a difference of 2 between the digits and expected that to give a change of 36.D I ignored the direction: swapping this number makes it smaller, not bigger.E I picked a number whose digits differ by 4 without noticing that the larger digit is already in front.3. D — 144 KA-0063 Divisibility rules: 2, 3, 4, 5, 9, 10
A I checked that it divides by 4 and stopped without testing the 9.B I checked the digit sum for 9 and stopped without testing the 4.C I saw an even number with a digit sum of 6 and treated 6 as good enough for 9.E I tested 9 with the digit sum and assumed any odd multiple of 9 also divides by 4.4. B — 1 KA-0089 Building and counting with unit cubes
A I assumed every small cube touches the outside somewhere.C I counted the cubes at the centre of each face, which are painted on one side.D I counted the corner cubes, which are the most painted rather than the least.E I gave the total number of small cubes instead of the unpainted ones.5. B — Bo only KA-0096 Truth-tellers and liars
A I took Ana's statement at face value without checking whether a truthful person could say it.C I decided nobody was lying without testing Ana's statement against that assumption.D I believed Ana's statement even after concluding she was a liar, which is what a liar's statement cannot be.E I gave up when Ana's statement looked circular, instead of testing one assumption all the way through.