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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.A drawer holds 10 red socks, 10 blue socks and 10 green socks, all mixed up in the dark. How many socks must you take out to be sure of having a matching pair?[5]
A2
B3
C4
D11
E31
3.A bag holds 5 black and 6 white stones. You repeatedly remove two stones: if they match you put a black one in, if they differ you put a white one in. What colour is the last stone?[5]
Ablack
Bwhite
Cit depends on the order of the moves
Dthe bag never gets down to one stone
Ewhite if the first two stones match
4.The numbers 1 to 9 are placed in a row in some order. Can every neighbouring pair add up to an odd number?[5]
AYes, and many orders work
BYes, but only one order works
CNo, there are too many odd numbers
DNo, nine places is an odd number of places
EOnly if the row starts with an even number
5.Ten different whole numbers, each bigger than zero, add up to 100. What is the largest that the biggest of them can be?[5]
A45
B50
C55
D91
E100
6.A room is 8 m long, 4 m wide and 2 m high. An ant walks across the floor and the walls from one bottom corner to the far top corner. What is the shortest distance in metres?[5]
Text description of the figure
The floor of a room, 8 metres by 4 metres, drawn flat, with one 8-metre-by-2-metre wall unfolded upwards from its far edge. A straight line joins the ant's starting corner on the floor to the opposite corner on the unfolded wall.
A9.2
B10
C11
D12
E14
7.A cube 3 units on each side is painted all over and then cut into unit cubes. How many of them have exactly two painted faces?[5]
A6
B8
C12
D24
E27
8.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
9.Can a 10 by 10 board be covered exactly by T-shaped tiles of four squares each, with no gaps and no overlaps?[5]
AYes, and it is straightforward
BYes, but the arrangement is fiddly
CNo, because 100 is not a multiple of 4
DNo, because the two colours cannot balance
EIt depends how the tiles are turned
10.A machine turns 3 into 7, 5 into 11 and 8 into 17. What does it turn 12 into?[5]
A19
B21
C24
D25
E29
11.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
12.You reach the ten 5-point questions with 18 minutes left. What is the best plan?[5]
AStart at the first and work through in order until time runs out
BRead all ten quickly, pick the two or three that look most doable, and fill in every answer before the end
CSpend the whole time on the last question, since it is the hardest
DAnswer the ones with the shortest statements
EDivide the time evenly, about 1.8 minutes each
13.How many squares of any size can be found on a 3 by 3 board?[5]
A9
B10
C12
D13
E14
14.On a die, opposite faces add to 7. A die rests on a table with 3 on top. What do the four side faces add up to?[5]
A10
B12
C14
D17
E18
15.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.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.
3.A — blackKA-0043Invariants and monovariants
BI noticed there are more white stones than black ones and guessed the majority colour would survive.
CI tried a few orders, saw different-looking positions along the way, and assumed the ending must vary too.
DI did not notice that every move takes two stones out and puts one back, so the count falls by exactly one each time.
EI assumed the first move settles the outcome instead of looking for a quantity that no move can change.
4.A — Yes, and many orders workKA-0049Parity of numbers (odd/even behavior)
BI found one arrangement that works and assumed a problem this fiddly could only have one answer.
CI saw five odd numbers against four even ones and called that a mismatch, when a row of nine needs exactly five of one and four of the other.
DI blamed the length of the row rather than checking how many odd and even numbers it has to hold.
EI decided the first number settles it without checking that starting even would need five even numbers, and only four exist.
5.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.
6.B — 10KA-0082Shortest paths on and around shapes
AI measured the straight line through the middle of the room, but the ant has to stay on the surfaces.
CI unfolded the room but folded out the short wall rather than the one that makes the shortest route.
DI chose an unfolding that gives a longer straight line and did not compare it with the others.
EI walked along the edges of the room, 8 then 4 then 2, instead of cutting straight across the surfaces.
7.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.
8.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.
9.D — No, because the two colours cannot balanceKA-0101Coloring arguments
AI checked that 100 divides by 4 and treated that as proof that a covering exists.
BI assumed a covering must exist somewhere and that I simply had not found it yet.
CI gave a reason that is not even true, since 100 really is a multiple of 4.
EI thought orientation could rescue it, but every turn of a T covers the same mixture of colours.
10.D — 25KA-0106Inferring a rule from examples
AI used the rule add 4, which fits the first example but not the others.
BI found a rule from two examples only and never tested it against the third.
CI doubled the input and forgot the extra one that every example needs.
EI used double and add five, which fits none of the examples exactly.
11.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.
12.B — Read all ten quickly, pick the two or three that look most doable, and fill in every answer before the endKA-0130Managing the 5-point block
AI worked in order, so a hard early question could eat the time meant for easier later ones.
CI assumed the hardest question is worth the most, but every question in this block scores the same.
DI used length as a proxy for difficulty, which is not reliable in this block.
EI spread the time equally, which is too little for the questions I could do and wasted on the ones I could not.
13.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.
14.C — 14KA-0139Dice and cube face relationships
AI subtracted only the top face from 21, forgetting the hidden bottom face as well.
BI guessed four middling numbers and added them without using the total of all six faces.
DI subtracted only the bottom face of 4 from the total of 21.
EI added the four largest numbers on the die rather than the four that are actually at the sides.
15.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.