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.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.

3.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.

4.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.

5.D — 6KA-0003Counting shapes hidden inside a figure

• AI counted only the three small triangles I could see as separate cells and stopped there.
• BI counted the three small triangles and added the whole big one, but forgot the two made of two cells.
• CI found the three small ones and one pair, but missed the second pair of neighbouring triangles.
• EI assumed every choice of two of the four lines would make a triangle, without checking each one.

6.C — 14KA-0009Counting cubes in a stack, including hidden ones

• AI counted only the cubes on the bottom layer and treated the rest as decoration.
• BI counted only the cubes whose faces I could actually see in the drawing.
• DI assumed every layer was a full 3 by 3 square and multiplied 9 by 2 for two layers.
• EI assumed the stack was a solid 3 by 3 by 3 block because it sits in a corner.

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 — 3KA-0068Tree diagrams

• AI read the two in exactly two heads as the answer rather than as a condition to count.
• CI counted the outcomes with two or more heads, so I included the three-head one as well.
• DI counted the orderings of three different objects instead of listing the eight outcomes.
• EI gave the total number of possible outcomes rather than the number with exactly two heads.

9.B — 15KA-0074Selections of a few objects

• AI doubled the group size instead of counting pairs.
• CI allowed a student to be paired with themselves, adding six impossible pairs.
• DI counted each pair twice, once in each order, and forgot to halve.
• EI multiplied 6 by 6, counting every ordered pair including a student with themselves.

10.A — 4KA-0154Inclusion–exclusion with two sets

• BI found how many students play at least one of the two and answered with that instead.
• CI added 18 and 15 to get 33, subtracted the 30 in the class, and used the 3 left over.
• DI removed the 7 from both groups and subtracted 11 and 8 from 30, taking the overlap away twice.
• EI answered with the number who play both, which is what the question gave me rather than what it asked for.

11.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.

12.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.

13.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.

14.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.

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.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.

17.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.

18.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.

19.A — 120KA-0166Counting up to symmetry

• BI counted every arrangement in a row, so each circular seating got counted once for every rotation.
• CI divided 720 by 20 or some other number instead of by the 6 rotations.
• DI divided by 12, treating reflections as identical too, though the question only allows rotations.
• EI fixed two people rather than one, dividing by an extra factor.

20.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.