Worksheet generator

1.C — 6KA-0025Counting up to symmetry

• AI also treated bracelets that are mirror images as the same, but only rotations were allowed.
• BI divided the number of beads by itself rather than dividing the number of arrangements by the number of rotations.
• DI divided the 24 arrangements by 2 instead of by the 4 rotations of a circle of four beads.
• EI counted every arrangement in a line and forgot that rotating a bracelet gives the same bracelet.

2.E — No, it is impossibleKA-0028Parity arguments

• AI found a sequence that turned over most of the cups and assumed the last one could be fixed somehow.
• BI matched the number of moves to the number of cups without checking whether the target is reachable at all.
• CI assumed that with enough moves any arrangement can be reached.
• DI thought the choice of which cups to flip could change whether the target is reachable.

3.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.

4.E — No, because the colours do not balanceKA-0034Coloring arguments

• AI assumed that because 14 is even, seven dominoes must fit somehow.
• BI tried a few arrangements, got most of the board covered, and assumed a full covering existed.
• CI claimed the right answer for the wrong reason, since 14 is in fact divisible by 2.
• DI thought removing a different pair of opposite corners could change the colour balance, but opposite corners always share a colour.

5.E — 9KA-0037Last-digit behavior of products and powers

• AI assumed a number this large has no reachable last digit and gave up on the cycle.
• BI found the cycle 7, 9, 3, 1 but used a remainder of 0 when the remainder is actually 2.
• CI found the cycle but counted its positions starting at zero, shifting my answer along by one.
• DI assumed the units digit of any power of 7 is always 7.

6.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.

7.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.

8.B — 6KA-0045Back-solve from the answer choices

• AI tested the smallest choice first and stopped at the first one that felt about right.
• CI read a third of itself as three times itself and solved the wrong equation.
• DI reversed the comparison, finding a number whose third is 4 less than itself in the wrong direction.
• EI picked the largest choice because dividing by three made me expect a big starting number.

9.D — 30KA-0047Counting by position (sliding window)

• AI counted only the sixteen smallest squares and stopped there.
• BI counted the small squares and the single big one, and forgot every size in between.
• CI counted the 2 by 2 squares as four rather than nine, splitting the board into blocks.
• EI used 4 by 4 positions for every size instead of shrinking the range as the square grows.

10.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.

11.B — 0.2KA-0052Area by subtraction (shaded regions)

• AI estimated by eye from the four corner pieces without comparing them to the whole square.
• CI used a diameter of 5 instead of a radius of 5 when working out the circle's area.
• DI assumed the circle covers half the square because it touches all four sides.
• EI found the fraction covered by the circle and gave that instead of the fraction outside it.

12.D — 12KA-0060Operation puzzles and cryptarithms

• AI added the digits of 132 instead of working out what A and B must be.
• BI found A + B correctly and then halved it, as if the question wanted one digit.
• CI spotted the 11 in the working and gave that instead of the sum it multiplies.
• EI found a pair of digits by trial and misadded them by one.

13.C — 48KA-0070Counting with restrictions

• AI treated the twins as one child and forgot they can also swap places with each other.
• BI glued the twins together into a single block but then forgot that the block can be arranged with the others in more ways than I counted.
• DI halved the 120 total, assuming the twins are together in exactly half the arrangements.
• EI counted every arrangement of five children and forgot the twins' condition entirely.

14.A — 12KA-0071Overcount, then correct

• BI divided the 120 arrangements by 6 rather than by the 10 movements that leave a bracelet looking the same.
• CI allowed for turning the bracelet but forgot it can also be flipped over.
• DI halved the 120 for flipping but forgot that turning also gives the same bracelet.
• EI counted every arrangement in a line, as if the bracelet had a fixed first bead.

15.B — 252KA-0075Complementary counting

• AI counted the numbers made entirely of digits other than 7 in every position, including a leading zero.
• CI counted the numbers with a 7 in each position separately and forgot that some were counted twice.
• DI added three lots of 90 and one extra hundred, double counting the 700s.
• EI counted the numbers with no 7 at all and gave that instead of subtracting it.

16.C — 7KA-0077The pigeonhole principle

• AI answered with the luckiest case, where the first three balls happen to match.
• BI found the worst case of two of each colour and forgot to take one more.
• DI multiplied the three colours by the three balls I need, rather than thinking about the worst case.
• EI worked from the number of balls of each colour rather than from the number of colours.

17.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.

18.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.

19.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.

20.B — 2KA-0094Weighing and balance puzzles

• AI assumed one weighing could separate nine possibilities, but it has only three outcomes.
• CI split the coins into halves each time, which wastes the balance's third outcome.
• DI weighed the coins one against another in pairs rather than in groups.
• EI compared each coin with a known good one in turn, which always works but is nowhere near the fewest.

21.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.

22.D — No, because the number of moves is oddKA-0097Parity arguments

• AI assumed enough moves can undo anything, without asking what each move preserves.
• BI tried a few sequences that nearly worked and assumed a better choice would finish the job.
• CI reached for the number of items rather than for what a single swap actually changes.
• EI thought the choice of swaps could change the outcome, when every swap has the same effect.

23.C — No, the last number is always evenKA-0098Invariants and monovariants

• AI found sequences ending in small numbers and assumed 1 was among the reachable ones.
• BI assumed a hard-looking question must have exactly one answer.
• DI found one sequence ending in 0 and assumed every sequence must end there.
• EI thought the order could change the outcome, when the quantity that decides it never changes.

24.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.

25.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.

26.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.

27.A — 1/10KA-0125Spot the trick vs. decide to grind

• BI cancelled the chain but stopped one factor early, leaving a 9 on the bottom.
• CI worked out the first bracket and assumed the rest made no difference.
• DI gave the last bracket on its own instead of the whole product.
• EI assumed everything cancels completely, leaving nothing behind.

28.B — Answer it with your best guess, mark it, and move onKA-0126Time triage: now, later, or guess

• AI let the time already spent decide, but that time is gone whatever I do next.
• CI moved on but left a blank, which scores nothing if I never get back to it.
• DI spent time re-reading work I had no reason to doubt, while 21 unseen questions waited.
• EI assumed the hardest questions are the best use of time, when the unseen easy ones are worth the same each.

29.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.