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Math Kangaroo practice

Name: ______________________Date: ____________5 questions · 25 points

1.On the street grid shown you may only walk right or down. One junction is closed. How many routes go from the top-left corner to the bottom-right corner?[5]
Text description of the figure
A grid of streets with junctions arranged 4 across and 4 down. The junction one step right and one step down from the top-left corner is marked closed with a cross.
- A8
- B10
- C12
- D18
- E20
2.One hundred apples are packed into twelve boxes. What is the largest number n for which you can always be sure that some box holds at least n apples?[5]
- A8
- B9
- C10
- D12
- E100
3.A class of 12 scored an average of 8. Another class of 18 scored an average of 13. What is the average for all 30 students together?[5]
- A10.5
- B11
- C11.5
- D12
- E21
4.How many squares of any size can be found on a 3 by 3 board?[5]
- A9
- B10
- C12
- D13
- E14
5.An 8 by 8 chessboard has two opposite corner squares removed, leaving 62 squares. Each domino covers exactly two squares that share an edge. Can the 62 squares be covered exactly by 31 dominoes?[5]
- ANo, because the two removed corners share a colour, so 32 squares of one colour remain and only 30 of the other
- BYes, because 62 is even and 31 dominoes cover exactly 62 squares
- CNo, because 62 is not divisible by 4
- DYes, but only if the dominoes may be placed diagonally
- ENo, because the board is no longer rectangular

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Answer key — Math Kangaroo practice

Each wrong choice carries the thinking that produces it. When a student picks C, this is what they were doing.

1.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.
2.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.
3.B — 11KA-0112Mixtures and weighted averages
- AI averaged the two averages, ignoring that the classes are different sizes.
- CI weighted the averages but used the wrong class size against each one.
- DI leaned towards the larger class's average without computing the totals.
- EI added the two averages together instead of combining them.
4.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.
5.A — No, because the two removed corners share a colour, so 32 squares of one colour remain and only 30 of the otherKA-0176Coloring arguments
- BI checked that the counts match but a matching count does not make a covering possible.
- CI invented a divisibility condition; dominoes cover two squares, so only divisibility by 2 could matter.
- DI changed the rules rather than testing them; the question says dominoes cover squares sharing an edge.
- EI appealed to the shape, but plenty of non-rectangular regions can be tiled by dominoes.

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