Practice guideClasses 3 to 8
Logical Reasoning for Olympiads: Rules and Practice
The logical reasoning section opens the SOF Maths and Science Olympiad papers, and school never teaches it. Here are the rules for each question type, then 15 practice questions with answers.
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15 practice questions at the end. Pick an answer for each question, then press Check my answers: your score and an explanation for every question appear at once. Nothing you choose leaves this page.
01Number and letter series
- Check the gaps 3, 7, 11, 15 goes up by 4 each time
- Look for multiplying 2, 6, 18, 54 is × 3 each time
- Squares and cubes 1, 4, 9, 16 are squares; 1, 8, 27, 64 are cubes
- Gaps that grow 2, 6, 12, 20 has gaps 4, 6, 8: the next gap is 10
- Two series in one 1, 10, 2, 20, 3, 30 alternates two patterns
- Letters number them A = 1 to Z = 26 and work with the numbers
- Opposite letters A and Z, B and Y: each pair adds up to 27
02Coding and decoding
- Shift codes CAT → DBU moves every letter one step forward
- Reverse codes the word is written backwards, then the rule is applied
- Number codes letters become their positions: BAD → 2 1 4
- Matching codes find a letter that appears in two coded words to fix its code
- Test your rule check it on every letter you are given before answering
03Directions
- Draw it sketch a compass with north at the top and east on the right
- Turns facing north, a right turn faces east and a left turn faces west
- Two right turns face the opposite way
- Shadows in the morning the Sun is in the east, so shadows fall to the west
- Straight-line distance if the walk makes a right angle, use Pythagoras: 3, 4, 5
04Ranking and order
- From the other end position = total − position from this end + 1
- Total in a row position from the left + position from the right − 1
- Comparisons write the names in a line, tallest to shortest, as you read
05Relations
- Family tree draw it: a line down for children, = for a married couple
- Grandfather's only son is the father
- Mother's only son is the speaker himself, if the speaker is a boy
- Mother's brother is the maternal uncle
06Analogies and odd one out
- Find the link first worker and tool, animal and home, young one, opposite, part and whole
- Odd one out test groups such as fruit or vegetable, prime or not, bird or mammal
- Check every option the link must work for all the others, not just two
07Clocks and calendars
- Angle between the hands |30H − 5.5M| degrees at H hours and M minutes
- Hands together 11 times in 12 hours
- Mirror image of a clock 11:60 minus the time shown: 2:30 looks like 9:30
- Odd days an ordinary year moves the weekday on by 1, a leap year by 2
- Leap years divisible by 4; century years only if divisible by 400
08Counting figures
- Squares in an n × n grid 12 + 22 + … + n2: 14 in a 3 × 3 grid
- Triangles count the small ones, then those made of two, three and more parts
- Painted cube cut into n × n × n 8 cubes have three painted faces, 12(n − 2) have two, 6(n − 2)2 have one and (n − 2)3 have none
- Mirror images left and right swap; water images :: top and bottom swap
09Practice questions1 mark each
1Find the next number: 5, 11, 23, 47, ___
2Find the next pair: AZ, BY, CX, ___
3If PEN is written as QFO, how is INK written?
4Ravi faces south and turns right twice. Which way is he facing now?
5In a row of 25 children, Tina is 9th from the left. What is her position from the right?
6Kabir is 7th from the left and 12th from the right in a row. How many children are in the row?
7Pointing to a girl, Arjun says, 'She is the daughter of my mother's only son.' How is the girl related to Arjun?
8Book is to library as painting is to ___
9Find the odd one out.
10What is the angle between the hands of a clock at 4:00?
11A clock seen in a mirror shows 2:30. What is the real time?
123 March is a Friday. What day is 24 March of the same year?
13How many squares of any size are there in a 3 × 3 grid?
14A 4 × 4 × 4 cube painted on every face is cut into 64 small cubes. How many small cubes have no painted face?
15Some cats are black. All black things are shiny. Which must be true?
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Answers and explanations
- 1DEach term is double the one before, plus 1: 47 × 2 + 1 = 95.
- 2AThe first letter goes forward from A and the second goes back from Z.
- 3BEvery letter moves one step forward: I → J, N → O, K → L.
- 4CSouth, turn right: west. Turn right again: north.
- 5D25 − 9 + 1 = 17.
- 6A7 + 12 − 1 = 18: Kabir is counted twice otherwise.
- 7BHis mother's only son is Arjun himself, so the girl is his daughter.
- 8CBooks are kept in a library; paintings are kept in a gallery.
- 9D3, 5 and 11 are prime numbers; 9 = 3 × 3.
- 10AEach hour mark is 30° apart: 4 × 30° = 120°.
- 11BSubtract from 11:60: 11:60 − 2:30 = 9:30.
- 12C24 − 3 = 21 days, exactly 3 weeks, so it is also a Friday.
- 13D9 small squares, 4 squares of 2 × 2 and 1 of 3 × 3: 14.
- 14AThe unpainted cubes form the inner 2 × 2 × 2 block: 8.
- 15BThe black cats are black things, so they are shiny. Nothing is known about the other cats.
How to use this guide
- Always draw: a compass for directions, a line for ranks, a tree for relations.
- Write the alphabet with its numbers (A = 1 to Z = 26) at the top of your rough sheet.
- Do ten reasoning questions a day for a month before the Olympiad.
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