Learn by Novus · Open practice pack v1

Diagrammatic reasoning: transformations, flows, and rule diagrams: open practice pack

100 untimed questions across 10 authored sections.

This is untimed educational practice for one cognitive skill. It is not a clinical, diagnostic, employment, or professionally recognized assessment, and it does not produce an IQ score or credential.

Learner worksheet

One hundred untimed self-check items across ten sections, moving from single-box flows to integrated multi-stage systems. Every diagram is described in words and every rule you need appears in the prompt. No outside knowledge required.

Choose one answer for each item. For a delayed-recall item, read its study text, cover it, and then answer without looking back.

Reading single-step flow diagrams

Each item describes one operation box. Apply its rule exactly as stated.

1. diagrammatic-reasoning-s01-q01

A flow diagram shows a single box labelled 'Add 4'. The number 9 enters the box. What number comes out?

  1. 5
  2. 12
  3. 13
  4. 36

Answer:

2. diagrammatic-reasoning-s01-q02

A box labelled 'Double' receives the number 7. What is the output?

  1. 9
  2. 14
  3. 21
  4. 49

Answer:

3. diagrammatic-reasoning-s01-q03

A box labelled 'Reverse' receives the letter sequence P-Q-R. What sequence comes out?

  1. P-Q-R
  2. Q-P-R
  3. R-Q-P
  4. R-P-Q

Answer:

4. diagrammatic-reasoning-s01-q04

A box labelled 'Remove the last item' receives the list: apple, pear, plum. What list comes out?

  1. apple, pear
  2. pear, plum
  3. apple, plum
  4. apple, pear, plum

Answer:

5. diagrammatic-reasoning-s01-q05

A box labelled 'Rotate 90° clockwise' receives a printed arrow pointing straight up. Which way does the arrow point on the way out?

  1. Up
  2. Down
  3. Left
  4. Right

Answer:

6. diagrammatic-reasoning-s01-q06

A box labelled 'Swap the first and last symbols' receives the sequence ★ ● ▲ ■. What comes out?

  1. ■ ● ▲ ★
  2. ● ★ ■ ▲
  3. ■ ▲ ● ★
  4. ★ ■ ▲ ●

Answer:

7. diagrammatic-reasoning-s01-q07

A box labelled 'Halve' receives the number 26. What is the output?

  1. 12
  2. 13
  3. 24
  4. 52

Answer:

8. diagrammatic-reasoning-s01-q08

A box labelled 'Toggle shading' turns white figures black and black figures white without changing the shape. A white triangle enters. What comes out?

  1. A white triangle
  2. A black square
  3. A white circle
  4. A black triangle

Answer:

9. diagrammatic-reasoning-s01-q09

A box labelled 'Repeat the whole sequence twice' receives the sequence A-B. What comes out?

  1. A-A-B-B
  2. A-B-A-B
  3. A-B-B-A
  4. B-A-B-A

Answer:

10. diagrammatic-reasoning-s01-q10

A box labelled 'Subtract 6' receives the number 6. What is the output?

  1. 0
  2. 1
  3. 6
  4. 12

Answer:

Two-step operator chains

Apply the two boxes strictly in the order shown, left to right.

11. diagrammatic-reasoning-s02-q01

The number 5 enters [Add 3] and the result enters [Double]. What is the output?

  1. 13
  2. 16
  3. 11
  4. 10

Answer:

12. diagrammatic-reasoning-s02-q02

The number 12 enters [Halve] and the result enters [Subtract 2]. What is the output?

  1. 4
  2. 5
  3. 8
  4. 10

Answer:

13. diagrammatic-reasoning-s02-q03

The letters C-A-T enter [Reverse] and the result enters [Remove the last letter]. What comes out?

  1. A-T
  2. C-A
  3. T-C
  4. T-A

Answer:

14. diagrammatic-reasoning-s02-q04

An arrow pointing north enters [Rotate 90° clockwise] and the result enters [Rotate 180°]. Which way does it finally point?

  1. North
  2. East
  3. West
  4. South

Answer:

15. diagrammatic-reasoning-s02-q05

The list 2, 4, 6, 8 enters [Remove the first item] and the result enters [Reverse]. What comes out?

  1. 8, 6, 4, 2
  2. 8, 6, 4
  3. 4, 6, 8
  4. 6, 4, 2

Answer:

16. diagrammatic-reasoning-s02-q06

The number 3 enters [Square] and the result enters [Add 1]. What is the output?

  1. 16
  2. 7
  3. 9
  4. 10

Answer:

17. diagrammatic-reasoning-s02-q07

A white circle enters [Toggle shading] and the result enters [Rotate 90° clockwise]. What comes out?

  1. A black circle
  2. A white circle
  3. A black square
  4. A white square

Answer:

18. diagrammatic-reasoning-s02-q08

The word STAR enters [Swap the first and last letters] and the result enters [Reverse]. What comes out?

  1. RATS
  2. RTAS
  3. SATR
  4. TARS

Answer:

19. diagrammatic-reasoning-s02-q09

The number 18 enters [Subtract 4] and the result enters [Halve]. What is the output?

  1. 9
  2. 5
  3. 7
  4. 14

Answer:

20. diagrammatic-reasoning-s02-q10

The number 6 enters [Add 2] then [Double], giving 16. If the same two boxes are wired in the opposite order, [Double] then [Add 2], what output does 6 produce?

  1. 16
  2. 14
  3. 12
  4. 10

Answer:

Inferring hidden rules from examples

Each box's rule is unknown: deduce it from the example pairs by testing every candidate against every pair.

21. diagrammatic-reasoning-s03-q01

A mystery box turns 3 into 6, 5 into 10, and 9 into 18. What is the box's rule?

  1. It adds 3
  2. It doubles
  3. It squares
  4. It adds 5

Answer:

22. diagrammatic-reasoning-s03-q02

A mystery box turns 10 into 7, 8 into 5, and 6 into 3. What is its rule?

  1. It halves
  2. It subtracts 2
  3. It subtracts 4
  4. It subtracts 3

Answer:

23. diagrammatic-reasoning-s03-q03

A mystery box turns CAT into TAC and STOP into POTS. What is its rule?

  1. It reverses all the letters
  2. It swaps only the first and last letters
  3. It moves the last letter to the front
  4. It writes the letters in alphabetical order

Answer:

24. diagrammatic-reasoning-s03-q04

A mystery box turns 4 into 16, 5 into 25, and 7 into 49. What is its rule?

  1. It multiplies by 4
  2. It adds 12
  3. It squares
  4. It multiplies by 5

Answer:

25. diagrammatic-reasoning-s03-q05

A mystery box turns the sequence 1-2-3 into 1-2-3-1, and the sequence X-Y into X-Y-X. What is its rule?

  1. It repeats the whole sequence
  2. It copies the first symbol onto the end
  3. It copies the last symbol onto the front
  4. It reverses the sequence

Answer:

26. diagrammatic-reasoning-s03-q06

A mystery box turns 7 into 15, 10 into 21, and 4 into 9. What is its rule?

  1. It doubles, then adds 1
  2. It adds 8
  3. It doubles, then subtracts 1
  4. It triples, then subtracts 6

Answer:

27. diagrammatic-reasoning-s03-q07

A mystery box turns a black square into a white square, and a white triangle into a black triangle. What is its rule?

  1. It turns every figure black
  2. It turns every figure white
  3. It changes the shape but keeps the shading
  4. It swaps the shading but keeps the shape

Answer:

28. diagrammatic-reasoning-s03-q08

A mystery box turns 12 into 6, 20 into 10, and 8 into 4. What is its rule?

  1. It subtracts 6
  2. It subtracts 10
  3. It halves
  4. It divides by 3

Answer:

29. diagrammatic-reasoning-s03-q09

A mystery box turns BOLT into BOL, RAIN into RAI, and TREE into TRE. What is its rule?

  1. It removes the last letter
  2. It removes the first letter
  3. It keeps only the first letter
  4. It removes all vowels

Answer:

30. diagrammatic-reasoning-s03-q10

A mystery box turns 2 into 8. Which rule is consistent with BOTH 2→8 and 3→27?

  1. It multiplies by 4
  2. It cubes
  3. It adds 6
  4. It squares, then adds 4

Answer:

Sequence and series patterns

Work out the pattern that generates each series, then predict the next term.

31. diagrammatic-reasoning-s04-q01

A series of frames shows: 1 dot, 2 dots, 3 dots, 4 dots. How many dots are in the next frame?

  1. 4 dots
  2. 6 dots
  3. 3 dots
  4. 5 dots

Answer:

32. diagrammatic-reasoning-s04-q02

A series of squares alternates: black, white, black, white. What colour is the next square?

  1. Black
  2. White
  3. Grey
  4. Striped

Answer:

33. diagrammatic-reasoning-s04-q03

An arrow rotates the same amount each frame: it points up, then right, then down. Where does it point next?

  1. Up
  2. Right
  3. Left
  4. Down

Answer:

34. diagrammatic-reasoning-s04-q04

A series of frames shows: 1 black dot, then 2 white dots, then 3 black dots, then 4 white dots. What does the next frame show?

  1. 5 white dots
  2. 5 black dots
  3. 4 black dots
  4. 6 white dots

Answer:

35. diagrammatic-reasoning-s04-q05

A series of shapes shows: triangle, square, pentagon. Which shape comes next?

  1. Circle
  2. Triangle
  3. Octagon
  4. Hexagon

Answer:

36. diagrammatic-reasoning-s04-q06

A star sits in the top-left corner of a square frame, then the top-right corner, then the bottom-right corner. Where is it in the next frame?

  1. Top-left corner
  2. Bottom-left corner
  3. Top-right corner
  4. The centre

Answer:

37. diagrammatic-reasoning-s04-q07

What is the next number in the series: 2, 4, 3, 6, 5, 10, 9, ...?

  1. 18
  2. 11
  3. 8
  4. 13

Answer:

38. diagrammatic-reasoning-s04-q08

What is the next letter in the series: A, C, F, J, ...?

  1. N
  2. M
  3. O
  4. P

Answer:

39. diagrammatic-reasoning-s04-q09

A series shows: a small arrow pointing up, a medium arrow pointing right, a large arrow pointing down. Sizes cycle small → medium → large → small, and the arrow keeps rotating 90° clockwise. What comes next?

  1. A large arrow pointing left
  2. A medium arrow pointing left
  3. A small arrow pointing up
  4. A small arrow pointing left

Answer:

40. diagrammatic-reasoning-s04-q10

A 3×3 grid of dot counts reads: row 1: 1, 2, 3; row 2: 2, 3, 4; row 3: 3, 4, ?. What number completes the grid?

  1. 4
  2. 6
  3. 5
  4. 7

Answer:

Branching flowcharts and decisions

Follow each decision diamond carefully. Answer the question asked at each branch before moving on.

41. diagrammatic-reasoning-s05-q01

A flowchart takes the number 7 to a decision: 'Is it even?' Yes leads to [Add 1]; No leads to [Add 3]. What is the output?

  1. 8
  2. 7
  3. 10
  4. 11

Answer:

42. diagrammatic-reasoning-s05-q02

A flowchart takes the number 12 to a decision: 'Is it greater than 10?' Yes leads to [Halve]; No leads to [Double]. What is the output?

  1. 6
  2. 24
  3. 12
  4. 5

Answer:

43. diagrammatic-reasoning-s05-q03

A flowchart takes the number 9 through two decisions. First: 'Is it even?' Yes → [Halve]; No → [Add 1]. Then: 'Is the result greater than 8?' Yes → [Subtract 5]; No → [Add 5]. What is the output?

  1. 15
  2. 5
  3. 10
  4. 4

Answer:

44. diagrammatic-reasoning-s05-q04

A flowchart takes the word ORANGE through two decisions. First: 'Does it have more than 5 letters?' Yes → [Remove the last letter]. Then: 'Does it start with a vowel?' Yes → [Reverse it]. What comes out?

  1. ORANG
  2. EGNARO
  3. ORANGE
  4. GNARO

Answer:

45. diagrammatic-reasoning-s05-q05

A flowchart takes the number 15 through two decisions. First: 'Is it a multiple of 3?' Yes → [Divide by 3]; No → [Add 3]. Then: 'Is the result odd?' Yes → [Add 1]; No → [Halve]. What is the output?

  1. 6
  2. 5
  3. 18
  4. 9

Answer:

46. diagrammatic-reasoning-s05-q06

A parcel-sorting flowchart asks in order: 'Does it weigh over 2 kg?' Yes → send to Depot B. No → 'Is it marked fragile?' Yes → send to Depot C; No → send to Depot A. Where does a 1.5 kg parcel marked fragile go?

  1. Depot A
  2. Depot B
  3. Depot C
  4. It cannot be determined

Answer:

47. diagrammatic-reasoning-s05-q07

Using the same sorting chart, over 2 kg → Depot B; otherwise fragile → Depot C; otherwise → Depot A, which parcel ends up at Depot A?

  1. A 3 kg parcel, not fragile
  2. A 1 kg parcel, marked fragile
  3. A 2.5 kg parcel, marked fragile
  4. A 1.8 kg parcel, not fragile

Answer:

48. diagrammatic-reasoning-s05-q08

A flowchart takes 24 through three decisions in order: 'Even?' Yes → [Halve]. 'Multiple of 3?' Yes → [Divide by 3]. 'Greater than 5?' No → [Add 10]. What is the output?

  1. 4
  2. 14
  3. 12
  4. 24

Answer:

49. diagrammatic-reasoning-s05-q09

A flowchart doubles even inputs and adds 5 to odd inputs. The output was 18. What was the input?

  1. 9
  2. 13
  3. 36
  4. 23

Answer:

50. diagrammatic-reasoning-s05-q10

A flowchart takes the pair (6, 9). First decision: 'Is the first number even?' Yes → [Swap the pair]. Second decision: 'Is the sum greater than 14?' Yes → output the sum; No → output the difference. What is the output?

  1. 3
  2. 6
  3. 9
  4. 15

Answer:

State and process diagrams

Track the current state after each signal. States change only when a listed transition allows it.

51. diagrammatic-reasoning-s06-q01

A traffic system cycles Green → Amber → Red → Green, one step per change. It currently shows Amber. What does it show after two changes?

  1. Green
  2. Red
  3. Amber
  4. Flashing amber

Answer:

52. diagrammatic-reasoning-s06-q02

A door has states Locked, Closed, and Open. 'Turn key' switches between Locked and Closed. 'Push' moves Closed to Open and has no effect otherwise. Starting from Locked, apply: turn key, push. What state results?

  1. Locked
  2. Closed
  3. Open
  4. It cannot be determined

Answer:

53. diagrammatic-reasoning-s06-q03

A ticket machine moves Idle → (insert coin) → Ready → (press button) → Printing → (finishes automatically) → Idle. From Idle, a coin is inserted and the button is pressed. What state is the machine in?

  1. Idle
  2. Ready
  3. Refunding
  4. Printing

Answer:

54. diagrammatic-reasoning-s06-q04

The same ticket machine ignores a button press while Idle. From Idle, the sequence is: press button, insert coin, press button. What state results?

  1. Idle
  2. Printing
  3. Ready
  4. The machine jams

Answer:

55. diagrammatic-reasoning-s06-q05

A lift starts at floor 2. Each U signal moves it up one floor and each D signal moves it down one. After the signals U, U, D, U, which floor is it on?

  1. Floor 3
  2. Floor 5
  3. Floor 4
  4. Floor 2

Answer:

56. diagrammatic-reasoning-s06-q06

A washing machine runs Fill → Wash → Rinse → Spin → Done. An 'advance' signal moves it one stage on; a 'hold' signal keeps the current stage. From Fill, the signals are: advance, advance, hold, advance. What stage is it in?

  1. Spin
  2. Rinse
  3. Wash
  4. Done

Answer:

57. diagrammatic-reasoning-s06-q07

A turnstile has two states. In Locked, a coin unlocks it and a push does nothing. In Unlocked, a push locks it again and a coin does nothing. From Locked, apply: coin, coin, push, push. What state results?

  1. Unlocked
  2. Locked
  3. Jammed
  4. It cannot be determined

Answer:

58. diagrammatic-reasoning-s06-q08

In a conveyor network, Station 1 sends even-numbered crates to Station 3 and odd-numbered crates to Station 2. Which stations does crate number 8 visit?

  1. Stations 1 and 2
  2. Station 2 only
  3. Stations 1, 2 and 3
  4. Stations 1 and 3

Answer:

59. diagrammatic-reasoning-s06-q09

A worker's status moves down one level (Fresh → Tired → Exhausted) after every second task completed at the current level. A rest moves the status up one level and resets the task count. Starting Fresh, the sequence is: task, task, task, rest, task. What is the final status?

  1. Fresh
  2. Tired
  3. Exhausted
  4. It cannot be determined

Answer:

60. diagrammatic-reasoning-s06-q10

A vending machine charges 30p. Credit accumulates as coins are inserted; the moment credit reaches 30p or more it vends and keeps only the excess as credit. Coins inserted: 20p, 20p, 10p. What is the final credit?

  1. 0p
  2. 10p
  3. 20p
  4. 30p

Answer:

Deducing unknown operators in pipelines

Work out what each mystery operator does from its examples before applying it to the new input.

61. diagrammatic-reasoning-s07-q01

Operator ◆ turns 4 into 8 and 10 into 20. What does ◆ give for the input 6?

  1. 10
  2. 12
  3. 8
  4. 36

Answer:

62. diagrammatic-reasoning-s07-q02

Operator ● turns A-B-C into C-B-A and P-Q into Q-P. What does ● give for R-S-T?

  1. R-S-T
  2. S-R-T
  3. R-T-S
  4. T-S-R

Answer:

63. diagrammatic-reasoning-s07-q03

Operator ◆ doubles its input. Operator ● turns 5 into 8 and 9 into 12. What is the output of 4 → [◆] → [●]?

  1. 11
  2. 14
  3. 10
  4. 19

Answer:

64. diagrammatic-reasoning-s07-q04

Operator ▲ turns 1-2-3 into 2-3, and 7-8-9-10 into 8-9-10. What does ▲ give for X-Y-Z?

  1. X-Y
  2. Z
  3. Y-Z
  4. X-Z

Answer:

65. diagrammatic-reasoning-s07-q05

A single mystery box turns 3 into 9 and 5 into 15. What is its rule?

  1. It squares
  2. It adds 6
  3. It multiplies by 3
  4. It adds 10

Answer:

66. diagrammatic-reasoning-s07-q06

In the pipeline input → [P] → [Q] → output, the input 2 produces 9 and the input 5 produces 15. P doubles its input. What must Q do?

  1. Add 3
  2. Double
  3. Subtract 5
  4. Add 5

Answer:

67. diagrammatic-reasoning-s07-q07

Operator ■ toggles shading (white ↔ black). Operator ▲ rotates a figure 90° clockwise. A white arrow pointing up passes through [▲] then [■]. What comes out?

  1. A white arrow pointing right
  2. A black arrow pointing right
  3. A black arrow pointing up
  4. A black arrow pointing left

Answer:

68. diagrammatic-reasoning-s07-q08

Two boxes, [Add 2] and [Halve], are wired one after the other in an unknown order. The input 6 produces the output 5. Which order are they in?

  1. Halve first, then Add 2
  2. Add 2 first, then Halve
  3. Either order gives 5
  4. Neither order can give 5

Answer:

69. diagrammatic-reasoning-s07-q09

In the pipeline input → [Reverse] → [◇] → output, the input 4-7-2 produces 2-7 and the input 1-5 produces 5. What does ◇ do?

  1. It removes the first item
  2. It keeps only the first item
  3. It reverses again
  4. It removes the last item

Answer:

70. diagrammatic-reasoning-s07-q10

In the pipeline 5 → [Double] → [X] → [Double] → 14, operator X applies the same rule every time. What does X give for the input 12?

  1. 9
  2. 15
  3. 24
  4. 6

Answer:

Loops and repeated processes

Repeat the rule exactly as stated and re-check the stopping condition after every pass.

71. diagrammatic-reasoning-s08-q01

Start with 3. Keep doubling until the number exceeds 20, then stop. What is the final number?

  1. 12
  2. 20
  3. 48
  4. 24

Answer:

72. diagrammatic-reasoning-s08-q02

Start with 2. Keep doubling until the number exceeds 30. How many doublings are performed?

  1. 5
  2. 4
  3. 3
  4. 6

Answer:

73. diagrammatic-reasoning-s08-q03

Start with 100. Repeatedly subtract 15 until the result is 40 or less. What is the final number?

  1. 40
  2. 25
  3. 55
  4. 45

Answer:

74. diagrammatic-reasoning-s08-q04

Start with 1. Add 3 on each pass and stop as soon as the total reaches 13 or more. How many passes run?

  1. 3
  2. 5
  3. 4
  4. 6

Answer:

75. diagrammatic-reasoning-s08-q05

Start with the word DIAGRAMS. Remove the last letter repeatedly until exactly 4 letters remain. What is left?

  1. DIAG
  2. DIAGR
  3. RAMS
  4. DIA

Answer:

76. diagrammatic-reasoning-s08-q06

A loop applies this rule: if the number is even, halve it; if odd, add 5. Start with 9 and apply the rule exactly 3 times. What results?

  1. 7
  2. 14
  3. 6
  4. 12

Answer:

77. diagrammatic-reasoning-s08-q07

Start with 5. On odd-numbered passes (1st, 3rd, ...) add 2; on even-numbered passes subtract 1. Run exactly 4 passes. What results?

  1. 9
  2. 6
  3. 7
  4. 8

Answer:

78. diagrammatic-reasoning-s08-q08

A loop halves the number while it is even. Start with 48. What is the final number?

  1. 6
  2. 3
  3. 1
  4. 24

Answer:

79. diagrammatic-reasoning-s08-q09

Start with 7. Loop: if the number is odd, add 3; if even, halve it. Stop when the number is below 5. What is the final number?

  1. 4
  2. 5
  3. 2
  4. 8

Answer:

80. diagrammatic-reasoning-s08-q10

A loop applies this rule: if the number is odd, add 2; stop when the number becomes even. Start with 9. What happens?

  1. It stops at 10
  2. It stops at 11
  3. It stops after exactly five passes
  4. It never stops

Answer:

Fault-finding in operator networks

A box may fail by passing its input through unchanged. Compare expected and actual outputs to isolate the fault.

81. diagrammatic-reasoning-s09-q01

The pipeline [Add 5] → [Double] receives 3. The expected output is 16, but the actual output is 8. Which box failed by passing its input unchanged?

  1. The Add 5 box
  2. The Double box
  3. Both boxes
  4. Neither box

Answer:

82. diagrammatic-reasoning-s09-q02

The pipeline [Reverse] → [Remove the last item] receives A-B-C-D. The expected output is D-C-B, but the actual output is A-B-C. Which box failed by passing its input unchanged?

  1. The Reverse box
  2. The Remove box
  3. Both boxes
  4. Neither box

Answer:

83. diagrammatic-reasoning-s09-q03

The pipeline [Subtract 2] → [Triple] receives 6. The expected output is 12, but the actual output is 18. Which box failed by passing its input unchanged?

  1. The Triple box
  2. Both boxes
  3. The Subtract 2 box
  4. Neither box

Answer:

84. diagrammatic-reasoning-s09-q04

The pipeline [Toggle shading] → [Rotate 180°] receives a white arrow pointing up. The expected output is a black arrow pointing down, but the actual output is a white arrow pointing down. Which box failed by passing its input unchanged?

  1. The Rotate box
  2. Both boxes
  3. Neither box
  4. The Toggle shading box

Answer:

85. diagrammatic-reasoning-s09-q05

The pipeline [Add 1] → [Double] → [Subtract 3] receives 4. The expected output is 7, but the actual output is 5. Exactly one box passed its input unchanged. Which one?

  1. The Add 1 box
  2. The Double box
  3. The Subtract 3 box
  4. It cannot be determined

Answer:

86. diagrammatic-reasoning-s09-q06

The pipeline [Double] → [Subtract 4] → [Double] receives 5. The expected output is 12, but the actual output is 20. Exactly one box passed its input unchanged. Which one?

  1. The first Double box
  2. The second Double box
  3. The Subtract 4 box
  4. None of the boxes

Answer:

87. diagrammatic-reasoning-s09-q07

A box labelled 'Add 10' receives 7 and outputs 27. Which fault mode best describes its behaviour?

  1. It did nothing
  2. It applied its rule twice
  3. It subtracted instead of adding
  4. It worked correctly

Answer:

88. diagrammatic-reasoning-s09-q08

A network splits the input 12 down two paths, top: [Halve]; bottom: [Subtract 8], then adds the two results. The expected output is 10, but the actual output is 16. Which component failed by passing its input unchanged?

  1. The Subtract 8 box
  2. Both boxes
  3. Neither box
  4. The Halve box

Answer:

89. diagrammatic-reasoning-s09-q09

The pipeline [Remove the first item] → [Reverse] → [Remove the first item] receives 1-2-3-4. The expected output is 3-2, but the actual output is 3-2-1. Exactly one box passed its input unchanged. Which one?

  1. The Reverse box
  2. The second Remove box
  3. None of the boxes
  4. The first Remove box

Answer:

90. diagrammatic-reasoning-s09-q10

A 'Double' box is tested twice. Test 1: input 3 gives output 6. Test 2: input 8 gives output 8. What is the soundest conclusion?

  1. It doubled correctly in the first test but passed the input unchanged in the second
  2. It always works correctly
  3. It never works correctly
  4. It halves even numbers

Answer:

Integrated system challenges

These combine rule deduction, ordering, loops, and faults. Verify each stage before moving to the next.

91. diagrammatic-reasoning-s10-q01

A white arrow pointing north passes through [Rotate 90° clockwise] → [Toggle shading] → [Rotate 90° clockwise]. What comes out?

  1. A white arrow pointing south
  2. A black arrow pointing north
  3. A black arrow pointing south
  4. A black arrow pointing west

Answer:

92. diagrammatic-reasoning-s10-q02

Operator ◆ doubles its input; operator ● adds a fixed number. The pipeline [◆] → [●] turns 6 into 15 and 10 into 23. What does the reversed pipeline [●] → [◆] give for the input 8?

  1. 19
  2. 16
  3. 13
  4. 22

Answer:

93. diagrammatic-reasoning-s10-q03

In a network, the numbers 3 and 5 enter a [Multiply] box. Its result and the number 4 then enter a [Difference] box that subtracts the smaller from the larger. What is the final output?

  1. 19
  2. 11
  3. 60
  4. 7

Answer:

94. diagrammatic-reasoning-s10-q04

A network routes by parity: even inputs go through [Square]; odd inputs go through [Add 7, then Halve]. The input 9 enters, and the result then passes through the network once more. What is the final output?

  1. 64
  2. 8
  3. 16
  4. 32

Answer:

95. diagrammatic-reasoning-s10-q05

Analogy: a white arrow pointing up is to a white arrow pointing right as a black arrow pointing left is to what?

  1. A black arrow pointing down
  2. A white arrow pointing up
  3. A black arrow pointing right
  4. A black arrow pointing up

Answer:

96. diagrammatic-reasoning-s10-q06

The pipeline [Add 2] → [Double] → [Subtract 1] should turn 5 into 13, but a test run produced 9. Exactly one box passes its input unchanged. What output will the same faulty machine give for the input 10?

  1. 23
  2. 21
  3. 19
  4. 20

Answer:

97. diagrammatic-reasoning-s10-q07

Start with 2. Each pass sends the value through [Double] then [Add 1]. Repeat passes until the value exceeds 20, then stop. What is the final value?

  1. 23
  2. 11
  3. 22
  4. 21

Answer:

98. diagrammatic-reasoning-s10-q08

A machine has a single store. When a number arrives: if the store is empty, the number is stored; otherwise the machine outputs the sum of the store and the arrival, then clears the store. The inputs are 3, 4, 6, 2 in that order. What is the second number the machine outputs?

  1. 7
  2. 8
  3. 6
  4. 9

Answer:

99. diagrammatic-reasoning-s10-q09

A figure series shows: (1 black dot, 8 white dots), (2 black, 6 white), (3 black, 4 white), (4 black, 2 white). How many dots in total does the fifth figure contain?

  1. 7
  2. 6
  3. 5
  4. 9

Answer:

100. diagrammatic-reasoning-s10-q10

Operator ★ turns 3 into 9 and 5 into 15. Operator ✦ turns 12 into 6 and 8 into 4. Starting with 4, apply ★, then ✦, then ★, then ✦. What is the final value?

  1. 12
  2. 9
  3. 18
  4. 6

Answer:

Authored hints

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This source contains no authored hints. No hint text was generated.

Answer key

ItemChoiceAnswer
diagrammatic-reasoning-s01-q01C13
diagrammatic-reasoning-s01-q02B14
diagrammatic-reasoning-s01-q03CR-Q-P
diagrammatic-reasoning-s01-q04Aapple, pear
diagrammatic-reasoning-s01-q05DRight
diagrammatic-reasoning-s01-q06A■ ● ▲ ★
diagrammatic-reasoning-s01-q07B13
diagrammatic-reasoning-s01-q08DA black triangle
diagrammatic-reasoning-s01-q09BA-B-A-B
diagrammatic-reasoning-s01-q10A0
diagrammatic-reasoning-s02-q01B16
diagrammatic-reasoning-s02-q02A4
diagrammatic-reasoning-s02-q03DT-A
diagrammatic-reasoning-s02-q04CWest
diagrammatic-reasoning-s02-q05B8, 6, 4
diagrammatic-reasoning-s02-q06D10
diagrammatic-reasoning-s02-q07AA black circle
diagrammatic-reasoning-s02-q08CSATR
diagrammatic-reasoning-s02-q09C7
diagrammatic-reasoning-s02-q10B14
diagrammatic-reasoning-s03-q01BIt doubles
diagrammatic-reasoning-s03-q02DIt subtracts 3
diagrammatic-reasoning-s03-q03AIt reverses all the letters
diagrammatic-reasoning-s03-q04CIt squares
diagrammatic-reasoning-s03-q05BIt copies the first symbol onto the end
diagrammatic-reasoning-s03-q06AIt doubles, then adds 1
diagrammatic-reasoning-s03-q07DIt swaps the shading but keeps the shape
diagrammatic-reasoning-s03-q08CIt halves
diagrammatic-reasoning-s03-q09AIt removes the last letter
diagrammatic-reasoning-s03-q10BIt cubes
diagrammatic-reasoning-s04-q01D5 dots
diagrammatic-reasoning-s04-q02ABlack
diagrammatic-reasoning-s04-q03CLeft
diagrammatic-reasoning-s04-q04B5 black dots
diagrammatic-reasoning-s04-q05DHexagon
diagrammatic-reasoning-s04-q06BBottom-left corner
diagrammatic-reasoning-s04-q07A18
diagrammatic-reasoning-s04-q08CO
diagrammatic-reasoning-s04-q09DA small arrow pointing left
diagrammatic-reasoning-s04-q10C5
diagrammatic-reasoning-s05-q01C10
diagrammatic-reasoning-s05-q02A6
diagrammatic-reasoning-s05-q03B5
diagrammatic-reasoning-s05-q04DGNARO
diagrammatic-reasoning-s05-q05A6
diagrammatic-reasoning-s05-q06CDepot C
diagrammatic-reasoning-s05-q07DA 1.8 kg parcel, not fragile
diagrammatic-reasoning-s05-q08B14
diagrammatic-reasoning-s05-q09B13
diagrammatic-reasoning-s05-q10D15
diagrammatic-reasoning-s06-q01AGreen
diagrammatic-reasoning-s06-q02COpen
diagrammatic-reasoning-s06-q03DPrinting
diagrammatic-reasoning-s06-q04BPrinting
diagrammatic-reasoning-s06-q05CFloor 4
diagrammatic-reasoning-s06-q06ASpin
diagrammatic-reasoning-s06-q07BLocked
diagrammatic-reasoning-s06-q08DStations 1 and 3
diagrammatic-reasoning-s06-q09AFresh
diagrammatic-reasoning-s06-q10C20p
diagrammatic-reasoning-s07-q01B12
diagrammatic-reasoning-s07-q02DT-S-R
diagrammatic-reasoning-s07-q03A11
diagrammatic-reasoning-s07-q04CY-Z
diagrammatic-reasoning-s07-q05CIt multiplies by 3
diagrammatic-reasoning-s07-q06DAdd 5
diagrammatic-reasoning-s07-q07BA black arrow pointing right
diagrammatic-reasoning-s07-q08AHalve first, then Add 2
diagrammatic-reasoning-s07-q09DIt removes the last item
diagrammatic-reasoning-s07-q10A9
diagrammatic-reasoning-s08-q01D24
diagrammatic-reasoning-s08-q02B4
diagrammatic-reasoning-s08-q03A40
diagrammatic-reasoning-s08-q04C4
diagrammatic-reasoning-s08-q05ADIAG
diagrammatic-reasoning-s08-q06D12
diagrammatic-reasoning-s08-q07C7
diagrammatic-reasoning-s08-q08B3
diagrammatic-reasoning-s08-q09A4
diagrammatic-reasoning-s08-q10DIt never stops
diagrammatic-reasoning-s09-q01BThe Double box
diagrammatic-reasoning-s09-q02AThe Reverse box
diagrammatic-reasoning-s09-q03CThe Subtract 2 box
diagrammatic-reasoning-s09-q04DThe Toggle shading box
diagrammatic-reasoning-s09-q05AThe Add 1 box
diagrammatic-reasoning-s09-q06CThe Subtract 4 box
diagrammatic-reasoning-s09-q07BIt applied its rule twice
diagrammatic-reasoning-s09-q08DThe Halve box
diagrammatic-reasoning-s09-q09DThe first Remove box
diagrammatic-reasoning-s09-q10AIt doubled correctly in the first test but passed the input unchanged in the second
diagrammatic-reasoning-s10-q01CA black arrow pointing south
diagrammatic-reasoning-s10-q02D22
diagrammatic-reasoning-s10-q03B11
diagrammatic-reasoning-s10-q04A64
diagrammatic-reasoning-s10-q05DA black arrow pointing up
diagrammatic-reasoning-s10-q06C19
diagrammatic-reasoning-s10-q07A23
diagrammatic-reasoning-s10-q08B8
diagrammatic-reasoning-s10-q09C5
diagrammatic-reasoning-s10-q10B9

Worked explanations

diagrammatic-reasoning-s01-q01

An 'Add 4' box outputs its input plus 4, so 9 becomes 9 + 4 = 13.

diagrammatic-reasoning-s01-q02

Doubling multiplies the input by 2, so 7 becomes 14.

diagrammatic-reasoning-s01-q03

Reversing writes the sequence back to front, so P-Q-R becomes R-Q-P. The middle item stays in place for a three-item sequence.

diagrammatic-reasoning-s01-q04

The box deletes only the final item (plum), leaving the rest in their original order.

diagrammatic-reasoning-s01-q05

A quarter turn clockwise moves an upward arrow to point right, imagine turning a clock hand from 12 to 3.

diagrammatic-reasoning-s01-q06

Only the two end symbols trade places (★ and ■); the middle symbols stay exactly where they were.

diagrammatic-reasoning-s01-q07

Halving divides the input by 2, so 26 becomes 13.

diagrammatic-reasoning-s01-q08

The box changes shading only, so the triangle keeps its shape but flips from white to black.

diagrammatic-reasoning-s01-q09

Repeating the whole sequence writes the complete run again after itself, giving A-B followed by A-B, not each symbol doubled in place.

diagrammatic-reasoning-s01-q10

6 − 6 = 0, a valid output; an operator can produce zero.

diagrammatic-reasoning-s02-q01

Work left to right: 5 + 3 = 8, then 8 × 2 = 16.

diagrammatic-reasoning-s02-q02

12 ÷ 2 = 6, then 6 − 2 = 4.

diagrammatic-reasoning-s02-q03

Reversing gives T-A-C; removing the last letter of that result leaves T-A. Apply each box to the previous box's output, not to the original input.

diagrammatic-reasoning-s02-q04

North turned 90° clockwise points east; a further 180° turn points it the opposite way, west.

diagrammatic-reasoning-s02-q05

Removing the first item leaves 4, 6, 8; reversing that gives 8, 6, 4.

diagrammatic-reasoning-s02-q06

3 squared is 9, and 9 + 1 = 10.

diagrammatic-reasoning-s02-q07

Toggling makes the circle black, and rotating a circle produces no visible change. Some operators have no effect on symmetric figures.

diagrammatic-reasoning-s02-q08

Swapping the end letters of STAR gives RTAS; reversing RTAS gives SATR. Track the intermediate result carefully at each stage.

diagrammatic-reasoning-s02-q09

18 − 4 = 14, then 14 ÷ 2 = 7.

diagrammatic-reasoning-s02-q10

Order matters in a pipeline: doubling first gives 12, then adding 2 gives 14, a different result from the original wiring.

diagrammatic-reasoning-s03-q01

Test each candidate against every pair: adding 3 fits 3→6 but fails 5→10, and squaring would turn 9 into 81. Only doubling fits all three examples.

diagrammatic-reasoning-s03-q02

Halving fits 6→3 but fails 10→7; subtracting 3 is the only rule that fits every pair.

diagrammatic-reasoning-s03-q03

For a three-letter word several rules look identical, so use the longer example to separate them: swapping only the end letters of STOP would give PTOS, not POTS. Full reversal fits both.

diagrammatic-reasoning-s03-q04

Multiplying by 4 fits 4→16 but fails 5→25, and adding 12 fails 5→25 too. Each output is the input multiplied by itself, so the box squares.

diagrammatic-reasoning-s03-q05

In both examples the output is the original sequence with its opening symbol appended at the end, nothing else moves.

diagrammatic-reasoning-s03-q06

Adding 8 fits 7→15 but fails 10→21, and tripling-minus-6 also fails 10→21. Doubling then adding 1 fits all three pairs. Always check every example, not just the first.

diagrammatic-reasoning-s03-q07

Both examples keep their shape while black and white exchange, so the rule acts on shading alone.

diagrammatic-reasoning-s03-q08

Subtracting 6 fits 12→6 but fails 20→10; each output is exactly half its input, so the box halves.

diagrammatic-reasoning-s03-q09

Every output is the input with just its final letter missing: removing vowels would have turned RAIN into RN.

diagrammatic-reasoning-s03-q10

All four candidate rules turn 2 into 8, but only cubing also turns 3 into 27: a single example rarely pins down a rule, so seek a second one before deciding.

diagrammatic-reasoning-s04-q01

The count rises by one dot per frame, so the next frame shows 5 dots.

diagrammatic-reasoning-s04-q02

The shading alternates every frame, and the last square shown was white, so black follows.

diagrammatic-reasoning-s04-q03

Each frame turns the arrow 90° clockwise (up → right → down), so the next quarter turn points it left.

diagrammatic-reasoning-s04-q04

Two features change independently: the count rises by one each frame while the shading alternates. After 4 white comes 5 black.

diagrammatic-reasoning-s04-q05

Each shape has one more side than the last (3, 4, 5), so a six-sided hexagon comes next.

diagrammatic-reasoning-s04-q06

The star moves one corner clockwise each frame, so from bottom-right it reaches bottom-left.

diagrammatic-reasoning-s04-q07

Two rules alternate: double, then subtract 1 (2→4→3→6→5→10→9). The next step doubles again: 9 × 2 = 18.

diagrammatic-reasoning-s04-q08

The gaps grow by one each step: +2, +3, +4, so the next gap is +5. Five letters past J is O.

diagrammatic-reasoning-s04-q09

Track each feature separately: the size cycle returns to small, while the rotation continues from down to left.

diagrammatic-reasoning-s04-q10

Each row rises by 1 from left to right (and each row starts one higher than the row above), so the final cell holds 5.

diagrammatic-reasoning-s05-q01

7 is odd, so the No branch applies: 7 + 3 = 10.

diagrammatic-reasoning-s05-q02

12 is greater than 10, so it is halved to 6.

diagrammatic-reasoning-s05-q03

9 is odd, so it becomes 10; 10 is greater than 8, so subtracting 5 gives 5. Re-test the condition on the updated value, not the original.

diagrammatic-reasoning-s05-q04

ORANGE has 6 letters, so it becomes ORANG; ORANG starts with the vowel O, so it is reversed to GNARO.

diagrammatic-reasoning-s05-q05

15 is a multiple of 3, giving 5; 5 is odd, so adding 1 gives 6.

diagrammatic-reasoning-s05-q06

At 1.5 kg the parcel passes the weight test, so the fragile question decides: fragile parcels go to Depot C.

diagrammatic-reasoning-s05-q07

Weight is tested first, so the 2.5 kg fragile parcel still goes to Depot B. Only a parcel at 2 kg or under that is not fragile reaches Depot A.

diagrammatic-reasoning-s05-q08

24 halves to 12; 12 divides by 3 to give 4; 4 is not greater than 5, so adding 10 gives 14.

diagrammatic-reasoning-s05-q09

Check each backwards path for consistency: an even input would be 9, but 9 is odd, contradiction. An odd input of 13 gives 13 + 5 = 18 and stays consistent.

diagrammatic-reasoning-s05-q10

6 is even, so the pair becomes (9, 6); the sum 15 exceeds 14, so the sum itself is output. Note that swapping never changes a sum.

diagrammatic-reasoning-s06-q01

From Amber, one change reaches Red and a second completes the cycle to Green.

diagrammatic-reasoning-s06-q02

Turning the key moves Locked to Closed, and pushing then opens the door.

diagrammatic-reasoning-s06-q03

The coin moves the machine to Ready and the button press moves it on to Printing.

diagrammatic-reasoning-s06-q04

The first press does nothing in Idle; the coin then moves it to Ready and the second press starts Printing. Signals with no listed transition are simply ignored.

diagrammatic-reasoning-s06-q05

Trace floor by floor: 2 → 3 → 4 → 3 → 4.

diagrammatic-reasoning-s06-q06

Fill advances to Wash, then Rinse; the hold keeps it at Rinse; the final advance reaches Spin.

diagrammatic-reasoning-s06-q07

The first coin unlocks it, the second coin is ignored, the first push relocks it, and the second push is ignored, ending Locked.

diagrammatic-reasoning-s06-q08

Every crate passes through Station 1 first; crate 8 is even, so it is routed on to Station 3 and never sees Station 2.

diagrammatic-reasoning-s06-q09

Two tasks drop the status to Tired; the third task only counts one of two; the rest restores Fresh and clears the count; the final task leaves the status still Fresh.

diagrammatic-reasoning-s06-q10

The second 20p lifts credit to 40p, triggering a vend that leaves 10p; the final 10p brings credit to 20p, below the vend threshold.

diagrammatic-reasoning-s07-q01

Both examples double their input, so ◆ turns 6 into 12.

diagrammatic-reasoning-s07-q02

Both examples reverse the sequence, so R-S-T becomes T-S-R.

diagrammatic-reasoning-s07-q03

● adds 3 in both examples, so the pipeline gives 4 × 2 = 8, then 8 + 3 = 11.

diagrammatic-reasoning-s07-q04

In both examples only the first item disappears, so ▲ removes the opening item and X-Y-Z becomes Y-Z.

diagrammatic-reasoning-s07-q05

Squaring fits 3→9 but fails 5→15, and adding 6 fails 5→15 too, only multiplying by 3 fits both examples.

diagrammatic-reasoning-s07-q06

P turns 2 into 4 and 5 into 10, so Q must turn 4 into 9 and 10 into 15, adding 5 fits both runs.

diagrammatic-reasoning-s07-q07

The rotation turns the arrow to point right, and the toggle then flips it from white to black.

diagrammatic-reasoning-s07-q08

Test both wirings: halving first gives 3 then 5, while adding 2 first gives 8 then 4, only halve-then-add matches the observed output.

diagrammatic-reasoning-s07-q09

Compute the intermediate stage first: reversing gives 2-7-4 and 5-1, so ◇ must drop the final item to leave 2-7 and 5.

diagrammatic-reasoning-s07-q10

Work backwards: the final Double must have received 7, so X turned 10 into 7: it subtracts 3, turning 12 into 9.

diagrammatic-reasoning-s08-q01

3 → 6 → 12 → 24; 24 is the first value over 20, and the loop stops there rather than doubling again.

diagrammatic-reasoning-s08-q02

2 → 4 → 8 → 16 → 32, four doublings bring the number past 30.

diagrammatic-reasoning-s08-q03

100 → 85 → 70 → 55 → 40; the condition '40 or less' is met exactly at 40, so no further pass runs.

diagrammatic-reasoning-s08-q04

1 → 4 → 7 → 10 → 13: four additions reach the threshold exactly.

diagrammatic-reasoning-s08-q05

DIAGRAMS has 8 letters, so four removals from the end leave the first four: DIAG.

diagrammatic-reasoning-s08-q06

9 is odd → 14; 14 is even → 7; 7 is odd → 12. Re-test the even/odd condition on each new value.

diagrammatic-reasoning-s08-q07

Track the pass number separately from the value: 5 → 7 → 6 → 8 → 7 after passes 1 to 4.

diagrammatic-reasoning-s08-q08

48 → 24 → 12 → 6 → 3; the loop stops at 3 because the condition 'while even' finally fails.

diagrammatic-reasoning-s08-q09

7 → 10 → 5 → 8 → 4. Note that 5 is not below 5, so the loop must continue past it. Read stopping conditions literally.

diagrammatic-reasoning-s08-q10

Adding 2 to an odd number always gives another odd number, so the stopping condition can never be met, checking whether a loop can terminate is part of reading a process diagram.

diagrammatic-reasoning-s09-q01

Simulate each fault: with Add 5 broken the output would be 6, and with both broken it would be 3. Only a broken Double explains 8, since 3 + 5 = 8 passed straight through.

diagrammatic-reasoning-s09-q02

If Reverse did nothing, the Remove box would trim A-B-C-D to A-B-C. Exactly what was observed. A broken Remove box would instead have produced D-C-B-A.

diagrammatic-reasoning-s09-q03

With Subtract 2 broken, the Triple box receives 6 unchanged and outputs 18, matching the observation. A broken Triple box would have given 4.

diagrammatic-reasoning-s09-q04

The rotation clearly happened (the arrow points down), but the shading never flipped, so the toggle box passed its input through untouched.

diagrammatic-reasoning-s09-q05

Simulate each single fault: a broken Add 1 gives 4 → 8 → 5; a broken Double gives 2; a broken Subtract 3 gives 10. Only the first matches the actual output of 5.

diagrammatic-reasoning-s09-q06

A broken first Double gives 5 → 1 → 2; a broken second Double gives 6; a broken Subtract 4 gives 10 → 10 → 20, which matches the observation.

diagrammatic-reasoning-s09-q07

Correct behaviour would give 17 and doing nothing would give 7; the observed 27 equals 7 + 10 + 10, the rule applied twice over.

diagrammatic-reasoning-s09-q08

With Halve broken the sum is 12 + 4 = 16, matching the observation; a broken Subtract 8 would give 6 + 12 = 18 and both broken would give 24.

diagrammatic-reasoning-s09-q09

With the first Remove broken, the full list is reversed to 4-3-2-1 and then trimmed to 3-2-1: the observed result. A broken Reverse would give 3-4, and a broken second Remove would give 4-3-2.

diagrammatic-reasoning-s09-q10

Judge each run separately against its expected output: 6 matches 3 doubled, but 8 does not match 16, an intermittent fault that only some test runs reveal.

diagrammatic-reasoning-s10-q01

Stage by stage: the arrow turns east, is toggled to black, then turns south. The two quarter turns combine into a half turn regardless of the toggle between them.

diagrammatic-reasoning-s10-q02

From the examples, ● adds 3 (12 → 15 and 20 → 23). Reversing the order changes the result: 8 + 3 = 11, then doubled to 22.

diagrammatic-reasoning-s10-q03

3 × 5 = 15, and the difference between 15 and 4 is 11.

diagrammatic-reasoning-s10-q04

First pass: 9 is odd, so (9 + 7) ÷ 2 = 8. Second pass: 8 is even, so it is squared to 64. Re-test the routing condition on the new value.

diagrammatic-reasoning-s10-q05

The first pair shows a 90° clockwise rotation with shading preserved; rotating a left-pointing arrow 90° clockwise points it up, still black.

diagrammatic-reasoning-s10-q06

Only a broken Add 2 explains the test: 5 → 10 → 9 (a broken Double gives 6 and a broken Subtract 1 gives 14). Through the same fault, 10 doubles to 20 and then loses 1, giving 19.

diagrammatic-reasoning-s10-q07

Each full pass maps n to 2n + 1: 2 → 5 → 11 → 23, and 23 is the first value past 20.

diagrammatic-reasoning-s10-q08

The machine pairs arrivals: 3 is stored, 4 triggers the output 7; then 6 is stored and 2 triggers the output 8.

diagrammatic-reasoning-s10-q09

Black dots rise by 1 while white dots fall by 2 each frame, so the fifth figure has 5 black and 0 white, 5 dots in total.

diagrammatic-reasoning-s10-q10

★ triples and ✦ halves, so the chain runs 4 → 12 → 6 → 18 → 9. Deduce each operator from its examples first, then trace the chain one step at a time.