Worksheet generator

1.C — 4KA-0020Paper folding and hole punching

• AI forgot that the punch goes through every layer at once, not just the top one.
• BI undid only one of the two folds before counting the holes.
• DI doubled for the first fold and then added two more rather than doubling a second time.
• EI counted a fold that was never made, doubling three times instead of twice.

2.A — 8KA-0031Counting paths on a grid

• BI assumed closing one junction removes about half the routes and halved the total of 20.
• CI counted the routes that pass through the closed junction and gave that instead of the ones that avoid it.
• DI subtracted only the 2 routes that reach the closed junction, not all the routes that continue through it.
• EI counted every route on the open grid and forgot to remove the ones through the closed junction.

3.C — 4KA-0040The pigeonhole principle

• AI answered with the lucky case, where the first two socks happen to match.
• BI used the number of colours as my answer without adding one for the sock that must repeat.
• DI worked from the number of socks of each colour instead of the number of colours.
• EI took the whole drawer, guaranteeing a pair but far more socks than are needed.

4.A — blackKA-0043Invariants and monovariants

• BI noticed there are more white stones than black ones and guessed the majority colour would survive.
• CI tried a few orders, saw different-looking positions along the way, and assumed the ending must vary too.
• DI did not notice that every move takes two stones out and puts one back, so the count falls by exactly one each time.
• EI assumed the first move settles the outcome instead of looking for a quantity that no move can change.

5.A — Yes, and many orders workKA-0049Parity of numbers (odd/even behavior)

• BI found one arrangement that works and assumed a problem this fiddly could only have one answer.
• CI saw five odd numbers against four even ones and called that a mismatch, when a row of nine needs exactly five of one and four of the other.
• DI blamed the length of the row rather than checking how many odd and even numbers it has to hold.
• EI decided the first number settles it without checking that starting even would need five even numbers, and only four exist.

6.C — 55KA-0078Extremal / worst-case counting

• AI found the smallest possible total of the other nine and gave that instead of what is left.
• BI assumed the biggest number could be at most half the total.
• DI made the other nine numbers all equal to 1, forgetting that they have to be different from each other.
• EI ignored the other nine numbers entirely, as if the biggest could take the whole total.

7.B — 10KA-0082Shortest paths on and around shapes

• AI measured the straight line through the middle of the room, but the ant has to stay on the surfaces.
• CI unfolded the room but folded out the short wall rather than the one that makes the shortest route.
• DI chose an unfolding that gives a longer straight line and did not compare it with the others.
• EI walked along the edges of the room, 8 then 4 then 2, instead of cutting straight across the surfaces.

8.C — 12KA-0090Counting cubes in a stack, including hidden ones

• AI counted the middle cube of each face, which has exactly one painted face rather than two.
• BI counted the corner cubes, which have three painted faces.
• DI counted every cube that is painted at all except the corners, without separating one face from two.
• EI gave the total number of small cubes rather than the ones with exactly two painted faces.

9.B — 9KA-0095The extremal principle

• AI divided 100 by 12 and rounded down instead of up.
• CI rounded the division to a convenient number without checking that a packing with a smaller maximum exists.
• DI gave the number of boxes as the answer rather than a number of apples.
• EI described the case where one box holds everything, which is possible but not guaranteed.

10.D — No, because the two colours cannot balanceKA-0101Coloring arguments

• AI checked that 100 divides by 4 and treated that as proof that a covering exists.
• BI assumed a covering must exist somewhere and that I simply had not found it yet.
• CI gave a reason that is not even true, since 100 really is a multiple of 4.
• EI thought orientation could rescue it, but every turn of a T covers the same mixture of colours.

11.D — 25KA-0106Inferring a rule from examples

• AI used the rule add 4, which fits the first example but not the others.
• BI found a rule from two examples only and never tested it against the third.
• CI doubled the input and forgot the extra one that every example needs.
• EI used double and add five, which fits none of the examples exactly.

12.B — 11KA-0112Mixtures and weighted averages

• AI averaged the two averages, ignoring that the classes are different sizes.
• CI weighted the averages but used the wrong class size against each one.
• DI leaned towards the larger class's average without computing the totals.
• EI added the two averages together instead of combining them.

13.B — Read all ten quickly, pick the two or three that look most doable, and fill in every answer before the endKA-0130Managing the 5-point block

• AI worked in order, so a hard early question could eat the time meant for easier later ones.
• CI assumed the hardest question is worth the most, but every question in this block scores the same.
• DI used length as a proxy for difficulty, which is not reliable in this block.
• EI spread the time equally, which is too little for the questions I could do and wasted on the ones I could not.

14.E — 14KA-0133Counting shapes hidden inside a figure

• AI counted the nine small squares and stopped, missing every larger one.
• BI counted the small squares and the whole board, forgetting the middle size entirely.
• CI found the 2 by 2 squares by splitting the board into blocks, getting two instead of four.
• DI slid the 2 by 2 square across and down but missed one of its four positions.

15.C — 14KA-0139Dice and cube face relationships

• AI subtracted only the top face from 21, forgetting the hidden bottom face as well.
• BI guessed four middling numbers and added them without using the total of all six faces.
• DI subtracted only the bottom face of 4 from the total of 21.
• EI added the four largest numbers on the die rather than the four that are actually at the sides.

16.B — 7KA-0157The pigeonhole principle

• AI thought that one more than the number of colours must give three of a kind, which only guarantees a PAIR.
• CI got to two of each colour, then added one more for each colour instead of one more in total.
• DI assumed the worst case meant emptying the largest pile of 12 reds first.
• EI answered with the number of socks I want rather than the number I must take to be sure of them.

17.B — 9KA-0158Last-digit behavior of products and powers

• AI used the first entry of the cycle because 2026 is even, without working out where in the cycle it lands.
• CI counted the cycle starting from zero, so a remainder of 2 put me one place too far along.
• DI assumed a power that large must end in 1.
• EI multiplied 3 by 2 and used the last digit of that.

18.A — 8KA-0175The pigeonhole principle

• BI used the number of possible remainders without adding one for the pair that must collide.
• CI doubled 7, thinking I needed two full sets of remainders.
• DI doubled 7 and added one, applying the pigeonhole idea to the wrong number of boxes.
• EI guessed from small cases without identifying what the boxes actually are.

19.A — No, because the two removed corners share a colour, so 32 squares of one colour remain and only 30 of the otherKA-0176Coloring arguments

• BI checked that the counts match but a matching count does not make a covering possible.
• CI invented a divisibility condition; dominoes cover two squares, so only divisibility by 2 could matter.
• DI changed the rules rather than testing them; the question says dominoes cover squares sharing an edge.
• EI appealed to the shape, but plenty of non-rectangular regions can be tiled by dominoes.

20.A — 54KA-0177Overcount, then correct

• BI counted every line joining two vertices and forgot to remove the 12 sides.
• CI counted each diagonal from both of its endpoints and forgot to halve.
• DI used 12 x 10 without halving, double counting every diagonal.
• EI subtracted 24 rather than 12, removing each side twice.