Worksheet generator

1.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.

2.D — 75KA-0023The multiplication principle

• AI added the number of options at each step instead of multiplying them.
• BI treated the two digits as one choice of ten rather than two independent choices of five.
• CI assumed the two digits had to be different, so I used 5 times 4 for them.
• EI used 5 choices for the letter position as well, forgetting there are only three letters.

3.D — 24KA-0073Arrangements of a few objects

• AI counted the letters rather than their arrangements.
• BI multiplied 4 by 3 and stopped, forgetting the last two positions.
• CI used 4 choices for every position, as if letters could repeat.
• EI used 4 choices in each of four positions, which allows every letter to repeat.

4.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.

5.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.

6.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.

7.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.

8.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.

9.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.

10.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.