Diagnostic test · 2026-09-07 · misses first. Each missed question is worked out together: every step draws itself and is read aloud, then the sample student answers a small check. The answer, the fastest method and the trap come at the end, and then a fresh question of the same skill proves it.
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Warm-up 3 marks · Money and time: cents, coins, clocks and elapsed time · Missed · answer given A (160) · correct answer C (80)after the steps
The first clock shows when a movie started and the second shows when it finished. How many minutes did it take?
Options: (A) 160 (B) 79 (C) 80 (D) 82 (E) 81
Work it out, one step at a time. 5 short steps, each with a check. The answer comes at the end.
Step 1 of 5
Read the start clock: short hand for the hour, long hand for the minutes.
What time did it start? Give the minutes past the hour.
A smaller question firstWhich big number does the long hand point at?
Right. Each big number is 5 minutes: count in 5s that many times. Then answer the step above.
= 40
Step 2 of 5
Read the finish clock the same way.
How many minutes past the hour does the finish clock show?
= 0
Step 3 of 5
Jump from 3:40 to the next full hour, 4:00.
How many minutes is that jump?
= 20
Step 4 of 5
Then from 4:00 to 5:00.
How many minutes is that?
= 60
Step 5 of 5
Add the two jumps.
What is 20 + 60?
= 80
Fastest method: Read both clocks, jump to the next full hour, count whole hours, add the last minutes.
The trap: treating an hour as 100 minutes, or subtracting the minute hands only.
Now a fresh one: the same skill, a new question, no hints, marked by code. "No hints" means the site gives none — only you can see whether anyone in the room helped.
Getting harder 4 marks · Working backwards and sharing in ratio · Missed · answer given A (35) · correct answer D (95)after the steps
4 tickets cost $19. How many dollars do 20 tickets cost?
Options: (A) 35 (B) 38 (C) 47 (D) 95 (E) 114
Work it out, one step at a time. 2 short steps, each with a check. The answer comes at the end.
Step 1 of 2
How many times bigger is 20 than 4? That is the scale factor, and it applies to both amounts.
What is 20 ÷ 4?
A smaller question firstWhat is 4 × 2?
Right. Now count on in 4s from there until you reach 20. Then answer the step above.
= 5
Step 2 of 2
Scale the other amount by the same factor.
What is 19 × 5?
A smaller question firstWhat is 10 × 5?
Right. Now work out 9 × 5 and add the two. Then answer the step above.
= 95
Fastest method: Find the scale factor between the two amounts of the same thing, then apply it to the other thing.
The trap: adding the difference instead of multiplying by the scale factor.
Now a fresh one: the same skill, a new question, no hints, marked by code. "No hints" means the site gives none — only you can see whether anyone in the room helped.
Challenge 7 marks · Number clues: digit puzzles and divisibility · Missed · answer given 192 · correct answer 183after the steps
What is the smallest three-digit number that is odd, a multiple of 3, contains the digit 8, has no digit 0, and has all different digits?
Work it out, one step at a time. 6 short steps, each with a check. The answer comes at the end.
Step 1 of 6
Do not start at the bottom of the range and test every number in turn. Start with the constraint that throws the most away — here that is 'contains the digit 8'.
Of the numbers still standing, how many survive 'contains the digit 8'?
A smaller question firstHow many numbers from 100 to 999 have no digit 8 at all?
Right. Take that away from all 900. Then answer the step above.
= 252
Step 2 of 6
Next strongest, onto the survivors only: 'a multiple of 3'.
Of the numbers still standing, how many survive 'a multiple of 3'?
= 85
Step 3 of 6
Next strongest, onto the survivors only: 'odd'.
Of the numbers still standing, how many survive 'odd'?
= 31
Step 4 of 6
Next strongest, onto the survivors only: 'has all different digits'.
Of the numbers still standing, how many survive 'has all different digits'?
= 28
Step 5 of 6
Last and weakest, on what little is left: 'has no digit 0'.
Of the numbers still standing, how many survive 'has no digit 0'?
= 26
Step 6 of 6
Everything left satisfies every constraint at once, so no more testing is needed — just read off the one the question asked for.
What is the smallest number left?
= 183
Fastest method: Apply the constraints in order of pruning power (divisibility / parity first) while keeping the leading digit as small as possible.
The trap: applying the weak constraints first, or dropping the distinct / no-zero filters.
Now a fresh one: the same skill, a new question, no hints, marked by code. "No hints" means the site gives none — only you can see whether anyone in the room helped.
Warm-up 3 marks · Shapes and area · Correct · answer given E (40) · correct answer E (40)
The diagram shows an L-shaped garden. Only two sides are labelled. What is the perimeter of the garden, in centimetres?
Options: (A) 39 (B) 42 (C) 20 (D) 41 (E) 40
Fastest method: Slide the short sides across: the perimeter equals a full rectangle's, 2 × (width + height).
The trap: trying to guess the unlabelled sides, or adding only the labelled ones.
Walk it through step by step (3 steps, each with a check)
Step 1 of 3
The two short unlabelled horizontal sides add up to the long top side, and the two vertical pieces on the right add up to the left side — slide them across.
So going all the way round is the same as twice 11 plus twice 9. What is 2 × 11?
A smaller question firstWhat is 10 × 2?
Right. Now work out 1 × 2 and add the two. Then answer the step above.
= 22
Step 2 of 3
And twice the height.
What is 2 × 9?
A smaller question firstWhat is 2 × 5?
Right. Now add 4 more lots of 2. Then answer the step above.
= 18
Step 3 of 3
Add them — the same as a full rectangle's perimeter.
What is 22 + 18?
A smaller question firstWhat is 22 + 10?
Right. Now add the last 8. Then answer the step above.
= 40
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Warm-up 3 marks · Space: cubes, nets, views and routes · Correct · answer given B (2) · correct answer B (2)
The net is folded to make a cube. Which number is on the face opposite the face numbered 5?
Options: (A) 3 (B) 2 (C) 0 (D) 1 (E) 4
Fastest method: In a net, faces two cells apart in a straight line are opposite; the two side flaps are opposite each other.
The trap: picking a face next to the given one — neighbours in the net are neighbours on the cube, never opposite.
Walk it through step by step (4 steps, each with a check)
Step 1 of 4
Faces that are two cells apart in the same straight line end up opposite each other when the net folds. In the column, the 1st and 3rd cells are a pair.
Which number is opposite 4?
= 7
Step 2 of 4
The 2nd and 4th cells of the column are the other pair.
Which number is opposite 9?
= 3
Step 3 of 4
The two side flaps fold up to become the left and right faces — opposite each other.
Which number is opposite 2?
= 5
Step 4 of 4
Now read off the pair that contains 5.
Which number is opposite 5?
= 2
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Warm-up 3 marks · Place value: building the biggest or smallest number · Correct · answer given C (297) · correct answer C (297)
Pat has the digit cards 6, 7, 9. Using every card exactly once she makes a single 3-digit number. What is the difference between the largest and smallest numbers she can make?
Options: (A) 315 (B) 387 (C) 297 (D) 288 (E) 306
Fastest method: Sort the digits into columns by place value — biggest digits to the biggest columns; the split into separate numbers is a distraction (the middle columns cancel in a difference).
The trap: tracking the whole numbers instead of the columns.
Walk it through step by step (4 steps, each with a check)
Step 1 of 4
Forget the digits for a second and look at the columns. A 3-digit number has 3 of them, and where a card lands is the only thing that decides what it is worth.
What is one card worth if you put it in the leading column of a 3-digit number?
A smaller question firstWhat is the leading column of a 2-digit number worth?
Right. One column further left is worth ten times as much. Then answer the step above.
= 100
Step 2 of 4
Biggest card into the biggest column, then straight down the line.
What is the largest number Pat can make?
A smaller question firstWhich card goes in the leading column?
Right. Now the biggest of the cards that are left goes next, and so on down the columns. Then answer the step above.
= 976
Step 3 of 4
Now run it backwards: smallest card into the biggest column.
What is the smallest number Pat can make?
A smaller question firstWhich card goes in the leading column?
Right. Now the smallest of the cards that are left goes next, and so on. Then answer the step above.
= 679
Step 4 of 4
The question never asked for either total on its own.
Largest minus smallest — what is the answer?
A smaller question firstWhat is 976 − 670?
Right. Now take away the last 9. Then answer the step above.
= 297
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Getting harder 4 marks · Data and chance: graphs, pictographs and spinners · Correct · answer given A (8) · correct answer A (8)
The pictograph shows the number of players for each child. Each full symbol stands for 4 players; a half symbol stands for 2. How many players were there for Ben?
Options: (A) 8 (B) 7 (C) 10 (D) 9 (E) 2
Fastest method: Multiply full symbols by the key, add half a key for a half symbol, then compare or add the rows.
The trap: counting symbols instead of what they stand for, or forgetting the half symbol.
Walk it through step by step (1 step, each with a check)
Step 1 of 1
Ben: 2 full symbols × 4.
How many players for Ben?
= 8
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Getting harder 4 marks · Counting the ways: cases and combinations · Correct · answer given B (31) · correct answer B (31)
Nia is exactly in the middle of a queue. Raf is 7 places behind Nia, and there are 8 people behind Raf. How many people are in the queue?
Options: (A) 29 (B) 31 (C) 16 (D) 30 (E) 32
Fastest method: The middle person has the same number behind as in front; count everyone behind the middle, double it, and add one for the middle person.
The trap: forgetting the middle person, or not doubling.
Walk it through step by step (3 steps, each with a check)
Step 1 of 3
Everyone behind Nia: 7 up to Raf, then 8 more behind Raf.
How many are behind Nia?
= 15
Step 2 of 3
Nia is exactly in the middle, so the same number are in front of her.
How many in front?
= 15
Step 3 of 3
Front + Nia + behind.
What is 15 + 1 + 15?
A smaller question firstWhat is 15 + 1?
Right. Now add 15 to that. Then answer the step above.
= 31
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Getting harder 4 marks · Shapes and area · Correct · answer given D (22) · correct answer D (22)
Lights are placed 4 m apart all the way around a triangular garden with sides 12 m, 40 m and 36 m, with a light at each corner. How many lights are needed?
Options: (A) 21 (B) 88 (C) 23 (D) 22 (E) 24
Fastest method: Posts around a closed shape: the number of posts equals the number of gaps, which is perimeter / gap.
The trap: adding one extra post as if the fence were a straight line.
Walk it through step by step (3 steps, each with a check)
Step 1 of 3
Lights go all the way around, so first find the whole distance around: add the sides.
What is 12 + 40 + 36?
A smaller question firstWhat is 12 + 40?
Right. Now add 36 to that. Then answer the step above.
= 88
Step 2 of 3
Lights are 4 m apart, so the number of gaps is 88 ÷ 4.
How many gaps?
A smaller question first4 × 20 = 80 makes part of 88. How many 4s are in the 8 left over?
Right. Now add that to the 20 you already have. Then answer the step above.
= 22
Step 3 of 3
Around a closed shape every light starts one gap and ends the previous one, so lights = gaps (no +1, unlike a straight row).
How many lights?
= 22
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Challenge 6 marks · Working backwards and sharing in ratio · Correct · answer given 60 · correct answer 60
Priya spends her prize money in three stages: first she spends 1/2 of it, then 1/5 of what is left, then 1/3 of what is left. She now has $16. How many dollars was the prize?
Fastest method: Reverse the chain — start at the end and undo each step with its inverse, last step first.
The trap: reporting an intermediate value of the reverse chain as the answer.
Walk it through step by step (3 steps, each with a check)
Step 1 of 3
Start from what is left and work backwards, undoing the last stage first. That stage took away one part in 3, so what came out of it is two-thirds of what went in. If two-thirds of an amount is known, the whole amount is that number divided by 2, then multiplied by 3.
How much was there just before the last stage?
A smaller question firstWhat is one part: 16 ÷ 2?
Right. What went in was 3 of those parts: multiply by 3. Then answer the step above.
= 24
Step 2 of 3
Now undo the stage before that, the same way. That stage took away one part in 5, so what came out of it is four-fifths of what went in. If four-fifths of an amount is known, the whole amount is that number divided by 4, then multiplied by 5.
How much was there just before stage 2?
A smaller question firstWhat is one part: 24 ÷ 4?
Right. What went in was 5 of those parts: multiply by 5. Then answer the step above.
= 30
Step 3 of 3
Now undo the stage before that, the same way. That stage took away one part in 2, so what came out of it is half of what went in. If half of an amount is known, the whole amount is that number divided by 1, then multiplied by 2. Undoing the first stage gives the amount at the very start — that is the question's answer, not any of the middle amounts.
How much was there just before the first stage?
= 60
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Challenge 9 marks · Patterns that repeat: cycles and step rules · Correct · answer given 360 · correct answer 360
At noon, 6 lights all flash at the same moment. After that they flash every 5, 4, 9, 8, 4 and 9 seconds respectively. How many seconds after noon do all 6 lights next flash together?
Fastest method: Each cycle repeats on its own period; they all line up at the LCM of the periods — build it from prime powers.
The trap: multiplying the periods together instead of taking the LCM.
Walk it through step by step (5 steps, each with a check)
Step 1 of 5
Two things that repeat every so often are together again at a common multiple of their two periods — and the FIRST time is the least common multiple.
After how many seconds are the first two lights back together?
= 20
Step 2 of 5
Bring in one more light. It is together with the group when the count is a multiple of the group's period AND of its own period — the least common multiple again.
After how many seconds are the first three lights back together?
= 180
Step 3 of 5
Bring in one more light. It is together with the group when the count is a multiple of the group's period AND of its own period — the least common multiple again.
After how many seconds are the first four lights back together?
A smaller question firstWhat is 180 × 8?
Right. They all divide into that, but it may not be the FIRST one: try the smaller multiples of 180 too. Then answer the step above.
= 360
Step 4 of 5
Bring in one more light. It is together with the group when the count is a multiple of the group's period AND of its own period — the least common multiple again.
After how many seconds are the first five lights back together?
= 360
Step 5 of 5
Bring in one more light. It is together with the group when the count is a multiple of the group's period AND of its own period — the least common multiple again.
After how many seconds are the first six lights back together?
= 360
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Challenge 10 marks · Counting the ways: cases and combinations · Not attempted · correct answer 148after the steps
Three different numbers are chosen from the list 3, 5, 7, ..., 105 and added together. How many different totals are possible?
Work it out, one step at a time. 3 short steps, each with a check. The answer comes at the end.
Step 1 of 3
First pin down the range. The smallest total uses the three smallest numbers.
What is the smallest possible total?
A smaller question firstWhat is 3 + 5?
Right. Now add 7 to that. Then answer the step above.
= 15
Step 2 of 3
And the largest total uses the three largest.
What is the largest possible total?
A smaller question firstWhat is 101 + 103?
Right. Now add 105 to that. Then answer the step above.
= 309
Step 3 of 3
Three odd numbers always add to an odd number, and swapping one chosen number for its neighbour moves the total by exactly two — so every odd value from the smallest total to the largest can be reached, and no even value can.
How many odd numbers are there from the smallest total to the largest, inclusive?
A smaller question firstWhat is 309 − 15?
Right. Halve it to count the jumps of 2, then add 1 for the first number. Then answer the step above.
= 148
Fastest method: Think in positions, not values: with an evenly spaced list, the three chosen positions control the total, and every position-sum from the smallest to the largest is achievable — so count the whole run of possible totals.
The trap: trying to list actual totals, or not checking that no total in the run is skipped.
Now a fresh one: the same skill, a new question, no hints, marked by code. "No hints" means the site gives none — only you can see whether anyone in the room helped.