30 problems — one per position, pulled from random authored years. Hints and solutions are locked until you submit. Retake as often as you want — every attempt is saved to your test history (if you're logged in).
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In the square you can see the digits from 1 to 9. A number is created by starting at the star, following the line and writing down the digits along the line while passing. For example, the line shown represents the number 42685. Which of the following lines represents the largest number?
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Answer: E
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Hint 1 of 2
Read off the digit string each path makes, then compare them as numbers.
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Hint 2 of 2
The biggest number starts with the largest leading digit; break ties by the next digit.
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Approach: trace each path into a number and compare
Each option traces a path from the star across cells of the 1-9 grid, writing the digit of every cell it passes.
Convert each path to its number and compare digit by digit from the left.
A staircase has 21 steps. Nick and Mike count the steps, one from bottom to top and the other from top to bottom. They meet at one step, which Nick counts as the 10th. Which number does Mike give this same step?
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Answer: C — the 11th
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Hint 1 of 2
They both count the very same step, just starting from opposite ends.
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Hint 2 of 2
On a 21-step staircase, think about how many steps sit below the meeting step and how many sit above it.
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Approach: count the same step from the other end
The two boys meet on one shared step on a staircase of 21 steps.
Mike counts that step from the top: the steps above it plus the meeting step itself make up his count, which lands on the 11th.
So Mike calls the meeting step the 11th — the answer is C.
Katrin has 38 matches. She uses all of them to make a triangle and a square that share no matches. Each side of the triangle is made of 6 matches. How many matches are in one side of the square?
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Answer: B — 5
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Hint 1 of 2
Work out how many matches the triangle uses, then see what is left for the square.
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Hint 2 of 2
Whatever the square gets, split it evenly across its four equal sides.
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Approach: subtract the triangle's matches, then divide the rest by 4
The triangle has three sides of 6 matches, so it uses 3 × 6 = 18 matches.
That leaves 38 − 18 = 20 matches for the square.
The square's four equal sides share these, so each side has 20 ÷ 4 = 5 matches.
A box can be moved only when nothing sits on it, so the order boxes become free is fixed by the starting picture.
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Hint 2 of 2
For each tower, check whether a box ends up under another box that was still trapped when it had to be placed; that ordering conflict makes a tower impossible.
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Approach: check the forced unloading order against each tower
From the start, only the top boxes are free; a box can be placed under another only if that other box was already moved.
Read each target tower from the bottom up and see whether every box placed below another could have been freed and set down before the box on top of it.
Tower C requires putting a box beneath one that was still pinned at that moment, which the rules forbid.
A rectangle is twice as long as it is wide. What fraction of the rectangle is shaded grey?
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Answer: B — \(\tfrac38\)
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Hint 1 of 2
Slice the rectangle along its diagonals and midline and compare the shaded triangles to the whole.
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Hint 2 of 2
Add up the grey pieces as a fraction of the full rectangle.
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Approach: split the rectangle into two equal squares and add the grey pieces
The rectangle is twice as long as wide, so it splits down the middle into two equal squares.
In the left square the grey is one triangle that is exactly half the square.
In the right square the grey triangle is half of a half, so a quarter of that square.
Grey is \(\tfrac12 + \tfrac14 = \tfrac34\) of one square, out of the two squares, so the shaded fraction is \(\tfrac34 \div 2 = \tfrac38\) — option (B).
A terrace is covered with square tiles of different sizes. The smallest tile has a perimeter of 80 cm. A snake lies along the edges of the tiles (see picture). How long is the snake?
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Answer: C — 420 cm
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Hint 1 of 3
A square's perimeter is four equal sides, so first turn the 80 cm into the length of one small-tile side.
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Hint 2 of 3
Use that small side as your ruler and walk along the snake, counting how many small sides fit into each straight stretch.
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Hint 3 of 3
Add up all the little side-lengths the snake covers, then multiply by the length of one side.
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Approach: find the small-tile side, then count how many of those lengths the snake covers
The smallest tile is a square with perimeter 80 cm, so each of its sides is 80 / 4 = 20 cm.
Reading the snake's path along the tile edges, it stretches across 21 of these small side-lengths.
A box-shaped water tank measures 4 m × 2 m × 1 m, and the water in it is 25 cm deep. The tank is then turned onto its side (see the picture on the right). How high is the water in the tank now?
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Answer: D — 1 m
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Hint 1 of 2
The amount of water does not change — only the shape of the space it fills.
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Hint 2 of 2
Find the water's volume, then divide by the area of the new bottom face to get the new height.
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Approach: the volume stays the same, so divide by the new base
The water's volume is 4 × 2 × 0.25 = 2 m³ (using 25 cm = 0.25 m).
After tipping, the tank rests on a 1 m × 2 m face, so the new bottom has area 2 m².
The two girls Eva and Olga and the three boys Adam, Isaac, and Urban play together with a ball. When a girl has the ball she throws it either to the other girl or to a boy. Every boy throws the ball only to another boy, but never back to the boy it just came from. The first throw is made by Eva to Adam. Who makes the 5th throw?
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Answer: A — Adam
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Hint 1 of 2
After Eva throws to Adam, the ball stays among the three boys, and a boy never throws back to whoever just threw to him.
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Hint 2 of 2
Track who holds the ball before each throw; the no-return rule on three boys forces the 5th thrower.
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Approach: trace the holder, using the no-immediate-return rule
Throw 1 is Eva to Adam, so Adam makes throw 2.
From then on the ball moves only among the three boys, each time to a boy other than the one who just passed it.
On a triangle of three boys this no-return rule sends the ball Adam -> (a boy) -> (the third boy) -> back to Adam.
In the last hockey game there were lots of goals. In the first half 6 goals were scored in total and the visiting team was leading. In the second half the home team scored another three goals and won the match. How many goals did the home team score in total?
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Answer: C — 5
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Hint 1 of 3
In the first half the two teams together scored 6, and the visitors were ahead.
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Hint 2 of 3
List the first-half scores where visitors lead: 6-0, 5-1 or 4-2.
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Hint 3 of 3
Then add the home team's 3 second-half goals and see which case lets them win.
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Approach: test first‑half splits that let home win
First half the goals add to 6 with visitors ahead, so the splits are visitors 6 home 0, visitors 5 home 1, or visitors 4 home 2.
The home team then scores 3 more; to win they need their total above the visitors' 6, 5, or 4.
Only 4-2 works: home ends with \(2 + 3 = 5\) against 4, a win, so the home team scored 5 in total, choice C.
You are given the three corner points of a triangle and want to add a fourth point to make the four corners of a parallelogram. In how many places can the fourth point be placed?
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Answer: C — 3
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Hint 1 of 2
Draw the triangle, then try sliding the new point off each of the three corners in turn.
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Hint 2 of 2
Each corner of the triangle can be the one that sits opposite the new point — count those choices.
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Approach: let each triangle corner be the one opposite the new point
Draw the triangle with corners A, B, C; the fourth point joins them into a parallelogram.
Pick which corner is opposite the new point: if it is A you get one parallelogram, if B another, if C a third.
That gives three different spots for the fourth point, so the answer is 3.
Instead of digits, Hannes uses the letters A, B, C, and D in a calculation. Different letters stand for different digits, and the addition ABC + CBA = DDDD is correct (each group of letters is a number written with those digits in order). Which digit does the letter B stand for?
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Answer: A — 0
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Hint 1 of 2
Add ABC and CBA column by column and notice the symmetry of the outer digits.
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Hint 2 of 2
The sum 101(A + C) + 20B must equal a four-digit repdigit DDDD; only D = 1 fits, which pins B.
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Approach: add by place value and match to a repdigit
Beatriz has five sisters whose ages are 2, 3, 5, 8, 10 and 17. Beatriz writes these ages in the circles of the diagram so that the sum of the ages in the four corners of the square equals the sum of the ages in the four circles in the horizontal row. What is this sum?
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Answer: D — 32
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Hint 1 of 2
The left and right circles belong to both groups, so when you set the two sums equal those two ages cancel out.
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Hint 2 of 2
That leaves top + bottom = centre + far-right; find the split of the six ages that makes this work, then add up the four corners.
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Approach: cancel the two shared circles, then place the rest
The corners (top, left, right, bottom) and the horizontal row (left, centre, right, far-right) share the left and right circles, so equal sums mean the unshared pairs match: top + bottom = centre + far-right.
Among 2, 3, 5, 8, 10, 17 the pairs 3 + 10 and 5 + 8 both make 13, so put 3 and 10 at top and bottom, and 5 and 8 at centre and far-right.
That leaves 2 and 17 for the shared left and right circles. Each sum is then (left + right) + (the matching 13) = (2 + 17) + 13 = 32.
In a pizzeria there is a basic pizza with tomato and cheese. It can be ordered with exactly one or exactly two of the following toppings: anchovies, artichokes, mushrooms or capers. The pizza comes in three sizes. How many different types of pizza are offered in total?
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Answer: A — 30
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Hint 1 of 2
Count the topping choices first: either pick one topping, or pick a pair of two toppings.
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Hint 2 of 2
Once you know how many topping choices there are, each one comes in 3 sizes.
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Approach: count topping choices, then multiply by sizes
Picking one topping: 4 ways. Picking a pair from the 4 toppings: the pairs are \(\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},\{3,4\}\), so 6 ways.
That is \(4 + 6 = 10\) topping choices.
Each choice comes in 3 sizes, so \(10 \times 3 = 30\) pizzas — the answer is A.
Starting from three numbers, the ‘addition machine’ makes three new ones by adding each pair together. For example, from {3, 4, 6} it makes {10, 9, 7}, and running it again gives {16, 17, 19}. We feed in the three numbers {20, 1, 3} and run the machine 2013 times. What is the biggest possible difference between two of the three resulting numbers?
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Answer: D — 19
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Hint 1 of 3
Don't follow the numbers themselves; watch the gaps between them.
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Hint 2 of 3
Work out the gaps in the example before and after one run of the machine and notice they are the same three gaps.
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Hint 3 of 3
If the gaps never change, the biggest gap at the end is the biggest gap you start with.
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Approach: watch the gaps between the numbers, not the numbers
Check the example: \(\{3,4,6\}\) has gaps 1, 2, 3, and after the machine \(\{7,9,10\}\) has gaps 2, 1, 3 — the very same three gaps, just shuffled.
So the three gaps between the numbers never change, no matter how many times you run the machine.
Starting from \(\{20,1,3\}\) the biggest gap is \(20 - 1 = 19\), and it is still 19 after 2013 runs, which is choice D.
A train has 12 carriages. In each carriage there is the same number of compartments. Mike is sitting in the 18th compartment behind the engine, this is in the 3rd carriage. Joanna is sitting in the 50th compartment behind the engine, this is in the 7th carriage. How many compartments are in one carriage?
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Answer: B — 8
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Hint 1 of 3
Every carriage holds the same number of compartments, so try the answer choices one at a time.
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Hint 2 of 3
If each carriage has 8 compartments, carriages 1 and 2 use compartments 1–16, so carriage 3 holds 17–24.
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Hint 3 of 3
Check that the same guess also lands compartment 50 inside carriage 7.
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Approach: test the answer choices: find how many compartments fit before each carriage
Try 8 compartments per carriage. Then carriages 1 and 2 fill compartments 1 through 16, so carriage 3 holds compartments 17 through 24 — and Mike's seat 18 lands there. ✓
With 8 each, carriages 1–6 fill compartments 1 through 48, so carriage 7 holds 49 through 56 — and Joanna's seat 50 lands there. ✓
Both facts work only with 8, so there are 8 compartments in one carriage.
A panel has 4 circles. When Lucy touches a circle, that circle and every circle touching it switch colour (white ↔ black), as shown. Starting with all circles white, at least how many circles must Lucy touch, one after another, to make all four black?
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Answer: C — 4
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Hint 1 of 2
Touching a circle flips it and its neighbours; each circle must end flipped an odd number of times.
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Hint 2 of 2
Find the smallest set of touches making every circle's flip-count odd.
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Approach: make every circle flip an odd number of times
Touching a circle toggles it and its neighbours; all four start white and must end black (odd flips each).
One or two touches cannot make all four flip an odd number of times.
The fewest touches achieving this is 4 (touching each circle once works).
An apple and an orange weigh as much as a pear and a peach. An apple and a pear weigh less than an orange and a peach, and a pear and an orange weigh less than an apple and a peach. Which of the pieces of fruit is the heaviest?
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Answer: C — peach
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Hint 1 of 2
Turn each sentence into an inequality between sums of two fruits.
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Hint 2 of 2
Combine the inequalities to rank the fruits and spot the heaviest.
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Approach: combine the weight inequalities
The balance apple + orange = pear + peach rearranges to peach = apple + (orange − pear).
The two 'weigh less' facts force pear to be lighter than both apple and orange, so orange − pear is positive.
Then peach = apple + (a positive amount) beats apple, and likewise peach beats orange, so the heaviest is peach.
Each shelf holds a total of 64 deciliters of apple juice. The bottles come in three different sizes: large, medium and small (see picture). How many deciliters of apple juice does a medium bottle contain?
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Answer: D — 10
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Hint 1 of 2
Let large, medium, small bottles hold L, M, S deciliters; each shelf's bottles total 64.
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Hint 2 of 2
Write one equation per shelf and solve the system for M.
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Approach: set up one equation per shelf and solve
Counting bottles, the three shelves give 3L + 4S = 64, 2L + 2M + 3S = 64, and 4M + 6S = 64.
Solving the system gives S = 4, L = 16 and M = 10.
A large cube has side-length 7 cm. On each of its 6 faces, the two diagonals are drawn in red. The large cube is then cut into small cubes with side-length 1 cm. How many small cubes will have at least one red line drawn on it?
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Answer: B — 62
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Hint 1 of 2
A red face-diagonal only marks the unit cubes it passes through on that face; count by face then remove double counts.
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Hint 2 of 2
Edge and corner cubes can be crossed by diagonals on more than one face — don't count them twice.
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Approach: count marked cubes per face, then correct overlaps
Each face is a 7×7 grid of little squares; the two diagonals run through 7 + 7 − 1 = 13 of them (the centre square is shared).
Six faces give 6 × 13 = 78, but cubes along the edges and corners get a red line on two faces and were counted twice — there are 16 such double-counts.
So the number of unit cubes with at least one red line is 78 − 16 = 62.
Vania has a sheet of paper divided into nine equal squares. She folds it as shown — first the horizontal folds, then the vertical folds — until the coloured square is on top of the stack. She wants to write the numbers 1 to 9, one per square, so that after folding they read in order from top to bottom, starting with 1 on top. On the unfolded sheet shown, which numbers should she write in places a, b and c?
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Answer: C — a = 7, b = 5, c = 3
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Hint 1 of 2
Track where each unfolded square ends up in the stack after the horizontal then vertical folds.
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Hint 2 of 2
Reverse the folds to read which numbers land at positions a, b and c on the flat sheet.
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Approach: reverse the fold order to map stack layers to grid cells
Folding horizontally then vertically stacks the nine squares; the coloured square is on top (number 1), and lower layers get 2,3,...
Unfolding to the flat sheet, each layer returns to its cell, spreading the numbers in a fixed pattern.
Reading positions a, b, c gives a = 7, b = 5, c = 3 - option C.
The map shows some islands connected by bridges. A navigator wants to visit each island exactly once. He started at Cang Island and wants to finish at Uru Island, and he has just reached the black island in the centre. In which direction must he go now to be able to complete his route?
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Answer: C — South.
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Hint 1 of 2
He has to visit every island exactly once, so the move he makes now must not strand any island he still needs to reach.
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Hint 2 of 2
If he picks a direction that walks him into a dead-end corner before the rest are visited, he can never get back; only one direction keeps a path open all the way to Uru.
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Approach: choose the only move that leaves a single-visit path to Uru
He must pass through each island once and finish at Uru, so from the centre he cannot step toward any group of islands he would later be unable to leave.
Going North, East or West leads him into a part of the map he would have to enter or leave twice, leaving some island unvisited.
Heading South is the one move that still lets him reach every remaining island exactly once and end at Uru.
