A mushroom grows a little bigger every day. Over five days Maria took a photo of this mushroom, but she put the photos in the wrong order (see picture). Which order of the photos shows the mushroom growing, from left to right?
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Answer: A — 2-5-3-1-4
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Hint 1 of 2
A mushroom only gets bigger from one day to the next, so order the photos from smallest cap to largest cap.
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Hint 2 of 2
Find the tiniest mushroom and read the labels in growing order.
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Approach: order the photos from smallest to largest mushroom
The mushroom grows every day, so the correct order goes from the smallest cap to the biggest, fully opened cap.
Reading the photo labels from smallest to largest gives the sequence in choice A: 2-5-3-1-4.
Eli drew a board on the floor with nine squares and wrote a number in each one, starting from 1 and adding 3 each time, until the board was full. Three of the numbers she wrote are shown in the picture. Which number below could be the one in the colored box?
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Answer: E — 22
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Hint 1 of 3
Skip-count out loud: start at 1 and keep adding 3 until you have written nine numbers.
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Hint 2 of 3
The colored box must hold one of the numbers Eli actually wrote, so check which answer choice shows up in your skip-count.
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Hint 3 of 3
Notice that some choices never appear in the count-by-3 list at all.
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Approach: write out Eli's count-by-3 list and see which choice fits
Eli starts at 1 and keeps adding 3, so the nine numbers she writes are 1, 4, 7, 10, 13, 16, 19, 22, 25.
Look at the choices: 14, 17 and 20 are never on this list, and 10 is already shown in another box, so they cannot sit in the colored box.
The only choice left that Eli really wrote is 22, so the colored box holds 22, which is choice E.
Paulo took a rectangular sheet of paper, yellow on one side and green on the other, and made the folds shown by the dotted lines to build a little paper plane. To decorate it, he punched one round hole, marked on the last picture. When he unfolded the sheet again, he found several holes in it. How many holes did he count?
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Answer: D — 8
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Hint 1 of 2
One punched hole passes through every layer the paper is folded into at that spot.
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Hint 2 of 2
Count how many layers stack up where the hole is punched.
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Approach: count layers pierced by the single punch
When the folded plane is punched, the hole goes through all the paper layers stacked there.
Unfolding spreads those holes out; the layers give 8 holes in total.
Three quarters painted means exactly 3 out of every 4 little squares are colored.
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Hint 2 of 2
Count colored squares versus total squares on each tray — one tray is off.
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Approach: check the painted fraction on each tray
For each tray, count the coloured squares and the total squares and compare with three quarters.
Trays A, B, D and E each have exactly 3 out of every 4 squares coloured, but tray C has only 5 of its 10 squares coloured (one half), so child C was wrong.
Gaspar has these seven different pieces, each made of equal little squares. He uses all the pieces to build rectangles with different shapes (and so different perimeters). How many different perimeters can he get?
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Answer: B — 3
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Hint 1 of 2
All the pieces together cover a fixed number of small squares — that's the area of every rectangle he makes.
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Hint 2 of 2
List the rectangle shapes with that area and count their different perimeters.
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Approach: fix the total area, then count distinct rectangle shapes
The pieces have 1, 2, 3, 4, 5, 6 and 7 squares, so together they always cover 1+2+3+4+5+6+7 = 28 squares.
A rectangle of 28 squares can be 1×28, 2×14 or 4×7, which give 3 different perimeters.
Cynthia paints each region of the figure in a single colour: red, blue or yellow. Regions that touch each other must be painted different colours. In how many different ways can Cynthia paint the figure?
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Answer: E — 6
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Hint 1 of 3
Find which regions actually share a border, since only touching regions are forced to be different colours.
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Hint 2 of 3
Colour one region first, then count how many free colour choices are left for each region next to it.
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Hint 3 of 3
Multiply the number of choices region by region as you work inward.
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Approach: colour the regions one at a time and multiply the free choices
Colour the outermost region first: there are 3 colours to pick from.
Each region just inside it touches the one outside, so it must use a different colour, which usually leaves only a small number of choices.
Multiplying the choices region by region as you move inward gives 6 different ways to paint the whole figure, choice E.
Denis ties his dog with an 11-metre rope, one metre away from a corner of a fence about 7 metres by 5 metres, as shown. Denis places 5 bones near the fence, as in the picture. How many bones can the dog reach?
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Answer: E — 5
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Hint 1 of 3
The dog is tied just outside the fence, so the rope has to hug the fence and bend around each corner to reach a bone.
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Hint 2 of 3
Walk the rope along the fence step by step, counting the metres (the tick marks) and bending it at every corner.
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Hint 3 of 3
Check how far 11 metres of rope can reach once it turns the corners.
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Approach: follow the rope along the fence, counting metres and bending at corners
The dog is tied 1 metre from the top-right corner, so the rope wraps along the top edge, down a side, and around the bottom, bending at each corner.
Counting the tick marks (each 1 metre), the farthest bone along this path is still within the 11 metres of rope once it bends around the corners.
So the rope is long enough to reach every bone, and the dog can grab all 5 of them, choice E.
Luana builds a fence using pieces of wood 2 metres long and half a metre wide, all the same shape. The picture shows the finished fence. How long is the fence, in metres?
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Answer: B — 6.5
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Hint 1 of 2
Each board is 2 m long, but where boards overlap, length is shared, not doubled.
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Hint 2 of 2
Add the board lengths along the fence, subtracting the overlaps.
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Approach: add board lengths along the fence, accounting for overlaps
The fence is made of 2 m boards whose ends overlap as shown in the picture.
Adding the lengths along the run, with the shared overlaps counted once, the total length is 6.5 m.
Amelia built a crown using 10 copies of the small piece shown. The pieces were joined so that touching sides always show the same number, as in the picture, where four pieces are filled in. What number appears in the coloured triangle?
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Answer: A — 1
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Hint 1 of 3
All ten pieces are exact copies, so the same little numbers repeat as you go around the ring.
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Hint 2 of 3
Where two pieces touch, the touching numbers match, which lines the pieces up the same way each time.
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Hint 3 of 3
Once you know how one piece sits, every piece sits the same, so jump that pattern around to the colored triangle.
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Approach: use the repeating pattern of identical pieces around the ring
Every piece is the same copy, and because touching sides must show equal numbers, each piece is placed in the very same way as its neighbour.
So the numbers repeat in the same order all the way around the crown, like a pattern that copies itself ten times.
Reading that repeating pattern from the filled-in pieces around to the colored triangle, the number landing there is 1, choice A.
Ana, Bia and Cris have, together, 100 reais. They go to the movies and each one pays her own entrance fee. Before paying, Ana had twice as much as each of her friends. After paying the fee, Ana now has three times what the other two friends have together. How much did the movie entrance cost?
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Answer: E — R$ 20
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Hint 1 of 3
Split the 100 reais into equal shares first: Ana's pile is as big as both friends' piles put together.
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Hint 2 of 3
After they pay, picture Ana's leftover money as three equal piles next to the two friends' leftover piles together.
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Hint 3 of 3
Compare how much Ana lost to how much the two friends lost in total.
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Approach: share the money into equal parts, then compare the leftovers in piles
Before paying, Ana has as much as both friends together, so split 100 into four equal piles of 25: Ana holds two piles (50) and each friend holds one pile (25).
After everyone pays the same ticket, Ana's leftover is three times the two friends' leftover together; so think of Ana's leftover as 3 small piles and the two friends' leftover together as 1 small pile, four small piles in all.
The four friends-and-Ana started with 100 reais and spent 3 tickets, and those leftovers split evenly: trying the choices, a 20-real ticket leaves Ana 30 and each friend 5 (the two friends have 10 together, and 30 is three times 10).
So the ticket cost 20 reais, choice E.
For older kids (algebra)Let each friend start with x, so Ana has 2x and 2x + x + x = 100 gives x = 25. After paying f each, 50 − f = 3((25 − f) + (25 − f)) leads to 5f = 100, so f = 20.
There are three flowers on the back of the left cactus. In total, the cactus on the right has six more flowers than the cactus on the left. How many flowers are on the back of the right cactus?
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Answer: D — 12
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Hint 1 of 2
Each cactus's total = the flowers you can see on its front + the flowers on its back; the left back is 3.
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Hint 2 of 2
Find the left cactus's total, add 6 for the right cactus, then subtract the right cactus's visible front.
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Approach: total each cactus, then subtract the visible front
The left cactus shows 5 flowers on its front plus 3 on its back, so its total is 8.
The right cactus has 6 more in total, that is 14, and it shows 2 flowers on its front, so its back has 14 − 2 = 12 flowers.
The points on opposite sides of an ordinary die add up to 7. This die is placed on the first square as shown and then rolled along, as in the picture, to the fifth square. When the die reaches the last square, what is the product of the numbers of points on its two coloured vertical faces?
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Answer: D — 18
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Hint 1 of 3
The two side faces are partners that always add to 7, so once you know one, you know the other.
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Hint 2 of 3
Roll a real die (or imagine one) tipping forward square by square and watch the side faces.
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Hint 3 of 3
Only the forward tips change the top and front faces; the two side numbers stay as a pair the whole way.
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Approach: roll the die step by step and read the two colored side faces at the end
Set a real die the same way as the picture and tip it forward, one square at a time, following the arrows to the last square.
Keep checking the two colored side faces, remembering that whatever shows on one side, its hidden partner is 7 minus that number.
On the last square the two colored side faces show 3 and 6, and 3 × 6 = 18, choice D.
Five friends decided to spend their vacation together. In a conversation, Adam said, “Yesterday was Wednesday.” Beto said, “Tomorrow will be Friday.” Carlos said, “The day before yesterday was Tuesday.” David said, “The day after tomorrow is Saturday.” Finally, Eli said, “Today is Monday.” One of them was wrong. Who was wrong?
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Answer: E — Eli
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Hint 1 of 2
Translate each statement into 'today is ___' and see if they agree.
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Hint 2 of 2
Four friends point to the same day; the odd one out is wrong.
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Approach: convert each clue to today's weekday
Adam (yesterday Wed), Beto (tomorrow Fri), Carlos (day-before-yesterday Tue) and David (day-after-tomorrow Sat) all mean today is Thursday.
Eli says today is Monday, disagreeing with the other four, so Eli was wrong.
The teacher wrote the numbers 1 to 8 on the board. Then he covered the numbers with triangles, squares and one circle (see picture). The sum of the numbers covered by the triangles equals the sum of the numbers covered by the squares, and the number covered by the circle is a quarter of that sum. What is the sum of the numbers covered by the triangles and the circle?
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Answer: C — 20
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Hint 1 of 3
First add up all the hidden numbers: 1 + 2 + 3 + ... + 8.
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Hint 2 of 3
The triangle pile and the square pile weigh the same, and the circle is just a small extra equal to a quarter of one of those piles.
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Hint 3 of 3
Try to split the total into two equal big piles plus a small piece that is a quarter of one big pile.
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Approach: split the total 36 into two equal piles plus a quarter-size circle
The hidden numbers are 1 through 8, and 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36.
The triangles and the squares make two equal piles, and the circle adds a quarter of one of those piles, so 36 splits as one pile + one equal pile + a quarter-pile.
That is the same as four-and-a-quarter quarter-piles making 36, so each quarter-pile is 4; one full pile (the triangles) is four of them, which is 16, and the circle is one quarter-pile, which is 4.
The triangles cover 16 and the circle covers 4, so together they cover 16 + 4 = 20, choice C.
Joana has several sheets of paper, each with a drawing of a parrot. She wants to paint only the head, tail and wing of the parrot, using red, blue or green. The head and tail may be the same colour, but the wing must not be the same colour as the head or the tail. How many sheets can she paint so that no two parrots are painted the same way?
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Answer: D — 12
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Hint 1 of 2
Pick colours for head, tail and wing in turn, remembering the wing's restriction.
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Hint 2 of 2
Split into 'head and tail same colour' versus 'head and tail different'.
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Approach: count by cases on the head/tail colours
If head and tail share a colour (3 ways), the wing has 2 allowed colours: 3×2 = 6.
If head and tail differ (3×2 = 6 ways), the wing must avoid both, leaving 1 choice: 6.
Jonas and Elias went to the beach for their vacation, where they had ice cream every day. Each ice cream they had, had two or three balls. On the last day of vacation, Jonas and Elias had had 23 and 19 ice cream balls in total, respectively. At least how many days were they on vacation?
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Answer: C — 8
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Hint 1 of 3
To use up the fewest days, give the bigger eater (Jonas, 23 balls) the most balls each day.
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Hint 2 of 3
Even eating 3 balls every single day, count how many days it takes to reach 23.
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Hint 3 of 3
The same number of days has to work for both boys, so check that 19 also fits in that many days.
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Approach: give the most balls per day to use the fewest days, then check both totals fit
To finish in as few days as possible, eat the biggest ice cream (3 balls) every day.
Counting by 3 toward Jonas's 23: seven days give only 21 balls, which is not enough, so they need at least 8 days.
Eight days really works for both: Jonas eats 3 balls on 7 days and 2 balls on 1 day (21 + 2 = 23), and Elias eats 3 balls on 3 days and 2 balls on 5 days (9 + 10 = 19).
So they were on vacation at least 8 days, choice C.
The Kangaroo Hotel has 30 floors, numbered 1 to 30, and each floor has 20 rooms, numbered 1 to 20. The code to enter a room is formed by writing the floor number followed by the room number, in that order. But a code can be confusing: for example, the code 111 could mean floor 11 room 1 or floor 1 room 11. Note that the code 101 is not confusing, since it can only mean floor 10 room 1 (floor 1 room 1 has the code 11, not 101). How many codes are confusing, including the one in the example?
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Answer: E — 18
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Hint 1 of 3
A code is confusing when the same digits can be cut into a floor and a room in two different correct ways.
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Hint 2 of 3
For three digits, you can cut after the first digit or after the second digit, so look for codes where both cuts give a real floor and a real room.
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Hint 3 of 3
Think about which three-digit codes start with a small floor but could also start with a teens floor.
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Approach: find three-digit codes that can be cut two ways, both giving a real floor and room
A three-digit code can be cut after the 1st digit (1-digit floor, 2-digit room) or after the 2nd digit (2-digit floor, 1-digit room); it is confusing when BOTH cuts give a real floor (1 to 30) and room (1 to 20).
For both cuts to work, the middle digit must be 1, so the codes look like floor-1-room, and they are exactly 11c and 21c where c is 1 to 9.
11c reads as floor 1 room 1c or floor 11 room c, and 21c reads as floor 2 room 1c or floor 21 room c, and c can be 1 to 9.
That is 9 codes of the form 11c and 9 codes of the form 21c, so 9 + 9 = 18 confusing codes, choice E.
