Worksheet generator

1.C — 4KA-0001Counting by position (sliding window)

• AI split the board into separate 2 by 2 blocks and only one whole block fitted, so I answered one.
• BI slid the square across the top row and found two positions, then forgot it can also move down.
• DI counted the four sliding positions and then added the two whole rows of the board as well.
• EI counted the nine small squares on the board instead of the 2 by 2 squares.

2.B — 28KA-0035Handshakes and pairs

• AI doubled the number of people, as if each person made two handshakes in total.
• CI let each person shake hands with all eight people including themselves before halving.
• DI multiplied 8 by 7 and forgot that each handshake was counted from both sides.
• EI multiplied 8 by 8, counting every ordered pair including a person with themselves.

3.C — 14KA-0051Complementary counting

• AI counted the multiples of 3 and gave that instead of the numbers that are not multiples.
• BI counted 21 as a multiple of 3 inside the range, so I removed one too many.
• DI removed only the multiples of 3 that are also even, so I subtracted three instead of six.
• EI gave the size of the whole range and forgot to remove anything at all.

4.C — 6KA-0054Counting shapes hidden inside a figure

• AI counted only the three small rectangles drawn as separate cells and stopped there.
• BI counted the three small rectangles and the whole one, and forgot the two made of two cells.
• DI counted a rectangle made of the left and right cells, even though they are not next to each other.
• EI assumed any pair of the four vertical lines makes a rectangle without checking each one.

5.C — 9KA-0072Systematic listing

• AI counted only the numbers with two identical digits, like 11, 22 and 33.
• BI required the two digits to be different, even though repeats are allowed.
• DI allowed a fourth digit that was not in the list.
• EI counted three-digit numbers instead of two-digit ones.

6.E — 8KA-0143Systematic listing

• AI counted the two available digits rather than the numbers they can build.
• BI counted the three positions instead of the numbers.
• CI used only two of the three positions when multiplying.
• DI multiplied 2 by 3, mixing the number of digits with the number of positions.

7.D — 9KA-0046Counting by position (sliding window)

• AI cut the board into four separate 2 by 2 blocks instead of sliding the square one step at a time.
• BI counted the three sliding positions along the top and doubled them for two rows.
• CI counted the positions along the top row and down the left column and added them.
• EI counted the sixteen small squares on the board rather than the 2 by 2 squares.

8.B — 4KA-0069Inclusion–exclusion with two sets

• AI added 18 and 14 without removing the overlap, then subtracted from 30.
• CI subtracted the 6 who play both twice instead of once.
• DI subtracted only the footballers from the class and forgot the chess players entirely.
• EI found how many play at least one sport and gave that instead of how many play neither.

9.A — 60KA-0172Selections of a few objects

• BI chose any 4 from all 9 people and ignored the two-and-two requirement.
• CI added the two counts instead of multiplying them.
• DI treated the choices as ordered, counting the same committee several times.
• EI used 5 x 4 x 2 or a similar shortcut rather than counting each selection properly.

10.A — 6KA-0173Counting up to symmetry

• BI counted every colouring of the four edges and forgot that rotations make some of them identical.
• CI divided 16 by 4, but that only works when no painting is left unchanged by a rotation, and some are.
• DI halved 16, treating only the 180 degree turn as a symmetry.
• EI listed by how many edges are black but forgot that two black edges can be adjacent or opposite.

11.C — 8KA-0012Casework

• AI counted each pair of digits once instead of counting both orders, such as 17 and 71.
• BI listed the pairs starting from 1 and 7 and forgot the number 80, where the second digit is zero.
• DI included 08 as a two-digit number, but a two-digit number cannot start with zero.
• EI counted both orders of every pair and then counted the pairs that reverse to themselves twice as well.

12.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.

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.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.

16.C — 4KA-0140The multiplication principle

• AI counted the codes that do start with A rather than the ones that do not.
• BI counted the letters available for the first position instead of counting whole codes.
• DI counted every arrangement of the three letters and forgot the restriction.
• EI allowed letters to repeat, which the words each exactly once rule out.

17.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.

18.A — 91/216KA-0168Complementary counting

• BI added 1/6 three times, which counts the overlapping cases more than once and would exceed 1 for seven rolls.
• CI worked out the probability of NO six and forgot to subtract it from 1.
• DI found the probability of three sixes instead of at least one.
• EI gave the probability for a single roll and ignored that there are three.

19.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.

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.