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1. A 3 by 3 board is drawn. How many 2 by 2 squares can be found on it, if they must line up with the grid?[3]
Text description of the figure A square board divided into a 3 by 3 arrangement of nine equal small squares.
2. Eight people meet and everyone shakes hands with everyone else exactly once. How many handshakes take place?[3]
3. How many of the numbers from 1 to 20 are not multiples of 3?[3]
4. A rectangle is divided by two vertical lines into three parts. How many rectangles of any size are in the picture?[3]
Text description of the figure A wide rectangle divided by two vertical lines into three smaller rectangles side by side.
5. A cafe offers 3 main courses and 2 puddings. How many different two-course meals can be ordered?[3]
M1 P1 P2 M2 P1 P2 M3 P1 P2 Text description of the figure A tree diagram. One starting point branches into three main courses, and each of those branches into two puddings, giving six paths in all.
6. How many different two-digit numbers can be written using only the digits 1, 2 and 3, if a digit may be used twice?[3]
7. How many three-digit numbers can be written using only the digits 1 and 2, with repeats allowed?[3]
Kangaroo Atlas · https://kangaroo-atlas.vercel.app · seed 2 · CC BY-NC-SA 4.0 · Independent project, not affiliated with Math Kangaroo in USA, NFP.
1. C — 4 KA-0001 Counting by position (sliding window)
A I split the board into separate 2 by 2 blocks and only one whole block fitted, so I answered one.B I slid the square across the top row and found two positions, then forgot it can also move down.D I counted the four sliding positions and then added the two whole rows of the board as well.E I counted the nine small squares on the board instead of the 2 by 2 squares.2. B — 28 KA-0035 Handshakes and pairs
A I doubled the number of people, as if each person made two handshakes in total.C I let each person shake hands with all eight people including themselves before halving.D I multiplied 8 by 7 and forgot that each handshake was counted from both sides.E I multiplied 8 by 8, counting every ordered pair including a person with themselves.3. C — 14 KA-0051 Complementary counting
A I counted the multiples of 3 and gave that instead of the numbers that are not multiples.B I counted 21 as a multiple of 3 inside the range, so I removed one too many.D I removed only the multiples of 3 that are also even, so I subtracted three instead of six.E I gave the size of the whole range and forgot to remove anything at all.4. C — 6 KA-0054 Counting shapes hidden inside a figure
A I counted only the three small rectangles drawn as separate cells and stopped there.B I counted the three small rectangles and the whole one, and forgot the two made of two cells.D I counted a rectangle made of the left and right cells, even though they are not next to each other.E I assumed any pair of the four vertical lines makes a rectangle without checking each one.5. C — 6 KA-0067 Tree diagrams
A I counted the puddings only and forgot that the main course is also a choice.B I added the number of mains to the number of puddings instead of multiplying them.D I used three puddings as well as three mains, without reading how many puddings there are.E I doubled the answer, as if the order of the two courses made a different meal.6. C — 9 KA-0072 Systematic listing
A I counted only the numbers with two identical digits, like 11, 22 and 33.B I required the two digits to be different, even though repeats are allowed.D I allowed a fourth digit that was not in the list.E I counted three-digit numbers instead of two-digit ones.7. E — 8 KA-0143 Systematic listing
A I counted the two available digits rather than the numbers they can build.B I counted the three positions instead of the numbers.C I used only two of the three positions when multiplying.D I multiplied 2 by 3, mixing the number of digits with the number of positions.