Worksheet generator

1.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.

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.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.

4.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.

5.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.

6.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.

7.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.

8.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.

9.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.

10.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.

11.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.

12.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.

13.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.

14.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.

15.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.

16.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.

17.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.

18.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.

19.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.

20.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.