About Logical Games
A logical game gives a group placed in some order - runners finishing a race, people living on the floors of a building - and a few statements about that order, and it asks about one position. The answer cannot be guessed and cannot be read off any single statement: it follows only by holding the possible orders in mind and striking out the ones the statements forbid. This topic covers the statement kinds that are used and the way to work through them.
What you need to understand
- Every one of these puzzles is the same task in different clothes: there is a set of possible orders, the statements cut the set down, and the question asks what is left.
- The statement kinds are few. Somebody is before or above somebody else; somebody is immediately before or above them; somebody is in a named position; somebody is at one of the two ends; somebody is next to somebody; somebody is not next to somebody; somebody is strictly between two others; and two people are an exact distance apart.
- The statements are not equally useful, and the order to use them in follows from that. A statement fixing somebody to a position, or putting them at an end, cuts the field hardest and is used first. A statement that gives an order cuts it roughly in half. A negative statement cuts least.
- A block is the most useful thing to build. Two people who are immediately next to each other can be treated as one object and slid along the row, which turns a puzzle of five into a puzzle of four.
- A distance statement is a block with a gap in it. Two people two places apart occupy three positions between them, and that object can be slid in the same way.
- A between statement fixes three people at once. It says nothing about which of the outer two comes first, so both orders have to be tried unless another statement decides it.
- Working in two passes is what makes these puzzles quick: put in what is fixed, then go back through the statements that could not be used yet, because a statement that was useless at the start often settles the puzzle once three positions are filled.
- A drawing is not optional. A row of numbered positions with the statements written underneath turns the puzzle into something that can be read rather than remembered.
- The answer to a question of this kind is not certain just because it fits. It has to be the only order that fits, and the habit worth forming is to read every statement back against the finished order at the end.
- When a question asks who is in a particular position, it is often quicker to work out who cannot be there. Ruling out four people answers the question as surely as placing one.
- Puzzles differ in what they place: people in a finishing order, people on floors, letters in a sequence, articles on shelves. The method does not change, and the drawing changes only in its labels.
- The direction of the sequence matters: floor 1 is the bottom and place 1 is the first to finish, and a statement about somebody being above somebody else has to be read with that in mind.
How to work through these questions
- Draw the positions and number them, then read every statement before writing anything. One of them usually fixes something, and starting there saves working through the others in the wrong order.
- Take the statements in order of strength: a named position, then an end, then a pair that must be adjacent, then an exact distance, then a plain order, and only then the negatives.
- Turn an adjacent pair into a block and write it as one object. Slide the block along and note where it cannot go: a block of two cannot start in the last position.
- For a between statement, mark the three people together and keep both possible internal orders until something rules one of them out.
- Work in two passes, because the second pass is where most of the puzzle is actually solved.
- Read every statement back against the finished order. If one is not satisfied, the order is wrong and there is no need to look for a second answer.
Mistakes that cost marks
- Reading before as immediately before. The first allows any distance and the second allows exactly one position, and they give different answers.
- Counting a block of two as being able to occupy five positions in a row of five. It can occupy four, because it cannot overhang the end.
- Forgetting that floor 1 is the bottom, so that a statement about somebody being above somebody else is read the wrong way round.
- Treating a between statement as though it also said which of the outer two comes first, when it does not.
- Spending the first minute on the negative statements, which cut the fewest possibilities and feel like progress without being any.
- Working the whole puzzle in the head and losing track of which order is which - the drawing is what keeps them apart.
- Answering about the wrong position after working out the whole row, having read place 1 where the question asked about place 5.
- Stopping at the first order that fits every statement, rather than checking that no other order does.
Worked example
Five runners finish a race. F finishes before D; E finishes immediately before D; A finishes between F and B; A finishes two places away from E; A finishes in place 2; and B finishes before F. Who finishes in place 5?
- Start with the statement that fixes something: A finishes in place 2. Everything else is measured from there.
- A finishes between F and B, so one of the two is in place 1 and the other is below A. B finishes before F, which decides it: B is in place 1 and F is below place 2.
- A finishes two places away from E. From place 2 that means E is in place 4, since place 0 does not exist.
- E finishes immediately before D, so D is in place 5.
- Read the statements back against the order, which is B, A, F, E, D: F before D holds, E immediately before D holds, A between F and B holds, A two places from E holds, A in place 2 holds, B before F holds. All six are satisfied, and no other order satisfies them, so the answer is D.
Answer: D finishes in place 5
Practice questions with answers
A few Logical Games questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
5 runners - F, B, E, A and D - finish a race, one in each of the first 5 places, numbered 1 to 5 in the order they finish.
- F finishes before D.
- E finishes immediately before D.
- A finishes between F and B.
- A finishes two places away from E.
- A finishes in place 2.
- B finishes before F.
Who finishes in place 5?
Answer: Option D — with explanation
A puzzle of this kind is settled by keeping the possible orders in mind and striking out the ones a statement forbids, and by starting with the statement that says the most.
A statement that fixes somebody to one place, or puts them at an end, cuts the field hardest and is the one to use first. A statement that puts two people immediately next to each other fixes them together as a block, which can then be slid along; one that puts somebody between two others fixes all three; and one that gives an exact distance between two people rules out everything except a few positions.
Working through these statements leaves one order and no other, and in it D is in place 5. The check worth doing at the end is to read each statement back against the order you have settled on - if one of them is not satisfied, the order is wrong and there is no need to look for a second answer.
Common mistakes
- A is somewhere else in the order: A
- B is somewhere else in the order: B
- F is somewhere else in the order: F
Question 2
6 people - A, B, E, F, C and D - live on the first 6 floors of a building, one to a floor, numbered 1 at the bottom.
- C lives above F.
- C lives between E and B.
- D does not live next to C.
- B does not live next to A.
- C lives between A and E.
- F lives between B and A.
- C lives above A.
- D does not live next to E.
- D lives between E and A.
- A lives two floors away from F.
Who lives on floor 5?
Answer: Option B — with explanation
A puzzle of this kind is settled by keeping the possible orders in mind and striking out the ones a statement forbids, and by starting with the statement that says the most.
A statement that fixes somebody to one place, or puts them at an end, cuts the field hardest and is the one to use first. A statement that puts two people immediately next to each other fixes them together as a block, which can then be slid along; one that puts somebody between two others fixes all three; and one that gives an exact distance between two people rules out everything except a few positions.
Working through these statements leaves one order and no other, and in it D is in place 5. The check worth doing at the end is to read each statement back against the order you have settled on - if one of them is not satisfied, the order is wrong and there is no need to look for a second answer.
Common mistakes
- F is somewhere else in the order: F
- E is somewhere else in the order: E
- A is somewhere else in the order: A
Question 3
5 runners - G, C, F, B and A - finish a race, one in each of the first 5 places, numbered 1 to 5 in the order they finish.
- A finishes between B and C.
- B finishes between G and F.
- G finishes between C and B.
- B finishes between F and G.
- B finishes between F and C.
- F is at one of the ends of the finishing order.
- C does not finish next to B.
- F finishes immediately before B.
- B finishes immediately before G.
Who finishes in place 5?
Answer: Option C — with explanation
A puzzle of this kind is settled by keeping the possible orders in mind and striking out the ones a statement forbids, and by starting with the statement that says the most.
A statement that fixes somebody to one place, or puts them at an end, cuts the field hardest and is the one to use first. A statement that puts two people immediately next to each other fixes them together as a block, which can then be slid along; one that puts somebody between two others fixes all three; and one that gives an exact distance between two people rules out everything except a few positions.
Working through these statements leaves one order and no other, and in it C is in place 5. The check worth doing at the end is to read each statement back against the order you have settled on - if one of them is not satisfied, the order is wrong and there is no need to look for a second answer.
Common mistakes
- G is somewhere else in the order: G
- F is somewhere else in the order: F
- A is somewhere else in the order: A
Question 4
6 runners - B, D, C, A, G and E - finish a race, one in each of the first 6 places, numbered 1 to 6 in the order they finish.
- A finishes between B and E.
- C finishes between B and G.
- E finishes between G and D.
- C finishes before B.
- C finishes immediately before B.
- G does not finish next to A.
- C finishes two places away from E.
Who finishes in place 4?
Answer: Option B — with explanation
A puzzle of this kind is settled by keeping the possible orders in mind and striking out the ones a statement forbids, and by starting with the statement that says the most.
A statement that fixes somebody to one place, or puts them at an end, cuts the field hardest and is the one to use first. A statement that puts two people immediately next to each other fixes them together as a block, which can then be slid along; one that puts somebody between two others fixes all three; and one that gives an exact distance between two people rules out everything except a few positions.
Working through these statements leaves one order and no other, and in it C is in place 4. The check worth doing at the end is to read each statement back against the order you have settled on - if one of them is not satisfied, the order is wrong and there is no need to look for a second answer.
Common mistakes
- D is somewhere else in the order: D
- B is somewhere else in the order: B
- A is somewhere else in the order: A
Question 5
5 runners - D, F, C, A and G - finish a race, one in each of the first 5 places, numbered 1 to 5 in the order they finish.
- G does not finish next to D.
- A finishes in place 4.
- C finishes before D.
- F finishes two places away from D.
- C finishes between F and A.
Who finishes in place 3?
Answer: Option A — with explanation
A puzzle of this kind is settled by keeping the possible orders in mind and striking out the ones a statement forbids, and by starting with the statement that says the most.
A statement that fixes somebody to one place, or puts them at an end, cuts the field hardest and is the one to use first. A statement that puts two people immediately next to each other fixes them together as a block, which can then be slid along; one that puts somebody between two others fixes all three; and one that gives an exact distance between two people rules out everything except a few positions.
Working through these statements leaves one order and no other, and in it D is in place 3. The check worth doing at the end is to read each statement back against the order you have settled on - if one of them is not satisfied, the order is wrong and there is no need to look for a second answer.
Common mistakes
- A is somewhere else in the order: A
- G is somewhere else in the order: G
- C is somewhere else in the order: C
Frequently asked questions
In what order should I use the statements?
Strongest first: a named position, then an end, then a pair that must be adjacent, then an exact distance, then a plain order, and the negatives last. A statement that halves the possibilities does more in ten seconds than three negatives do in a minute.
What is the quickest way to handle a block?
Write it as one object and count the positions it can start in. A block of two in a row of five can start in four places, not five, because it cannot overhang the end. Sliding the block and noting where it cannot go usually places it outright.
Does a between statement say which of the two outer people comes first?
No. It says only that the named person is somewhere between the other two. Both internal orders have to be kept open until another statement, or the answer, settles which one it is.
How do I know when the answer is certain?
When the statements leave only one order possible. That is stronger than checking that your order fits, because a second order might fit too. The safe habit is to read every statement back against the finished row at the end.
Do I need to draw anything?
Almost everyone who is quick at these puzzles does. A row of numbered positions with the statements underneath turns the puzzle into something that can be read rather than remembered, and it is the remembering that costs the time.
How long should one of these take?
About a minute and a half for five people and six statements, using a drawing from the start. The commonest way to lose time is to work without a drawing and to go back to the beginning when two orders get confused.
