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Logic Puzzles: Use Clues to Rule Things Out
Solve who-has-what puzzles with a grid of checks and crosses, put people in order from clues, and decide when a statement must be true and when you simply cannot tell.
The idea
In a logic puzzle, the clues never tell you everything at once. Your job is to use each clue to rule something out, then combine what is left. When only one possibility remains, you know it must be true. This kind of thinking is called deduction.
A logic grid helps. List the people down the side and the choices across the top. When a clue says something is true, put a check (✓) in that box and an ✗ in the rest of that row and column, because each choice belongs to only one person. When a clue says something is NOT true, put an ✗. Look for a row or column with only one empty box left.
Careful thinkers also know when they cannot be sure. "If it rains, recess is indoors" does not mean "If recess is indoors, it rained". Recess could be indoors for another reason, such as extreme cold. When the clues allow more than one answer, the honest answer is "I cannot tell".
When you finish, check your answer against every clue. A good puzzle has exactly one answer that fits them all. Coders do the same thing when they test a program: try the answer and see if any rule is broken.
Worked examples
Example 1 · Use a logic grid
Ana, Ben and Cara each joined a different club: chess, art or choir. Clue 1: Ana is not in choir. Clue 2: Ben is in art. Who is in which club?
- Clue 2: put ✓ for Ben–art. Put ✗ for Ben–chess and Ben–choir, and ✗ for Ana–art and Cara–art.
- Clue 1: put ✗ for Ana–choir. Ana's row now has only chess left, so Ana–chess ✓.
- Chess and art are taken, so Cara must be in choir.
- Check: Ana is not in choir ✓. Ben is in art ✓.
Ana: chess · Ben: art · Cara: choir
Example 2 · Put people in order
Dev, Eli, Fay and Gus use lockers 1 to 4 in a row. Fay uses locker 1. Gus's locker is right next to Fay's. Dev's locker number is smaller than Eli's. Who uses each locker?
- Fay is 1. The only locker next to 1 is 2, so Gus is 2.
- Dev and Eli are left with 3 and 4. Dev's number is smaller, so Dev is 3 and Eli is 4.
- Check every clue against the answer.
1 Fay · 2 Gus · 3 Dev · 4 Eli
Example 3 · Know when you cannot tell
Pat, Quinn and Rosa each brought one item: a soccer ball, a skipping rope or a frisbee. The only clue: Pat did not bring the skipping rope. Who brought what?
- Pat could have the soccer ball or the frisbee.
- If Pat has the soccer ball, Quinn and Rosa could share the other two in either order. The same is true if Pat has the frisbee.
- That is 4 possible answers. One clue is not enough.
You cannot tell. The puzzle needs more clues.
Common mistakes
Why this fails: Each choice belongs to only one person. A ✓ for Ben–art means no one else has art and Ben has nothing else.
Why this fails: "All top-shelf books are mysteries" does not mean every mystery is on the top shelf. Check which direction the rule goes.
Why this fails: If two answers fit every clue, the correct response is 'cannot tell', not a lucky guess.
Why this fails: A quick check catches a box you filled in too early. Test the answer clue by clue, like testing a program.
Foundation
Foundation practice
Decide what must be true, what cannot be true and when you cannot tell.
1. All the books on the top shelf are mysteries. This book is on the top shelf. Is it a mystery?
- Yes
- No
- Cannot tell
2. All the books on the top shelf are mysteries. This book is a mystery. Is it on the top shelf?
- Yes
- No
- Cannot tell
3. Mia, Noah and Omar each chose a different fruit: apple, banana or pear. Noah did not choose the apple. Omar chose the pear. Which fruit did Mia choose?
- banana
- apple
- pear
- Cannot tell
4. If it rains, recess is indoors. Today recess was outdoors. Did it rain?
- Yes
- No
- Cannot tell
5. If it rains, recess is indoors. Today recess was indoors. Did it rain?
- Yes
- No
- Cannot tell
Practice
Practice practice
Use a grid or a number line. Check your answer against every clue.
6. Lena, Raj and Sofia each do one activity: swimming, hockey or piano. Raj's activity does not happen in water. Sofia's activity does not use skates or water. Who does each activity?
7. Ada, Ben, Chloe and Dan each like a different colour: red, blue, green or yellow. Ada likes neither red nor blue. Ben likes blue. Chloe likes neither green nor red. Which colour does Dan like?
- green
- yellow
- red
- blue
8. Eva, Finn, Gia and Hugo stand in the lunch line. Gia is first. Hugo is right behind Eva. Finn is not last. What is the order from first to last?
- Gia, Eva, Hugo, Finn
- Gia, Finn, Eva, Hugo
- Gia, Hugo, Eva, Finn
- Finn, Gia, Eva, Hugo
9. Ivy is taller than Jay. Kai is shorter than Jay. Lu is taller than Ivy. Order them from tallest to shortest.
10. Ada, Ben, Chloe and Dan each like a different colour: red, blue, green or yellow. Clue 1: Ben likes blue. Clue 2: Ada does not like blue. Clue 3: Chloe likes yellow. Clue 4: Ada does not like red. Which clue could you remove and still solve the puzzle?
- Clue 1
- Clue 2
- Clue 3
- Clue 4
11. Every student in the chess club is also in the math club. Tara is not in the math club. Which statement must be true?
- Tara is in the chess club.
- Tara does not like math.
- Everyone in the math club plays chess.
- Tara is not in the chess club.
Think
Think practice
Find missing clues, test statements and work backward.
12. Pat, Quinn and Rosa each brought one item: a soccer ball, a skipping rope or a frisbee. Pat did not bring the skipping rope. Which ONE extra clue would give exactly one answer?
- Quinn has the skipping rope.
- Pat does not have the skipping rope.
- Rosa has the frisbee.
- Rosa does not have the soccer ball.
13. One of three friends ate the last cookie. Ana says, "Ben did it." Ben says, "I didn't." Cy says, "I didn't." Exactly one of them is telling the truth. Who ate the cookie?
- Ana
- Ben
- Cy
- Cannot tell
14. I am an odd two-digit number between 30 and 70. My digits add up to 11. My tens digit is smaller than my ones digit. What number am I?
15. Five runners finished a race: Ama, Bo, Cal, Di and Ed. Bo finished right after Ama. Cal finished before Ama. Di finished last. Ed finished before Cal. What was the finishing order?
Challenge
Challenge practice
Solve a bigger grid, then write a clue of your own.
16. Wes, Xia, Yuri and Zoe each borrowed a different book: an atlas, a comic, a dictionary or a poem book. Xia chose neither the atlas nor the comic. Wes and Yuri did not choose the dictionary. Zoe chose the comic. Yuri did not choose the poem book. Who borrowed each book?
17. Ella, Femi and Gwen each sit in a different seat: 1, 2 or 3. The only clue is: Femi is not in seat 3. That allows 4 answers. Write one more clue that leaves exactly one answer, and show the answer.
Answer key
- 1. Yes
The rule covers every book on the top shelf, and this book is one of them. - 2. Cannot tell
The rule says top-shelf books are mysteries. It does not say every mystery is on the top shelf. This one could be anywhere. - 3. apple
Omar has the pear. Noah cannot have the apple, so Noah has the banana. Mia has the apple. - 4. No
If it had rained, recess would have been indoors. It was outdoors, so it did not rain. - 5. Cannot tell
Rain is one reason for indoor recess, but there could be others, such as extreme cold. The rule does not work backward. - 6. Lena: swimming · Raj: hockey · Sofia: piano
Sofia cannot swim or play hockey, so she plays piano. Raj cannot swim, and piano is taken, so he plays hockey. Lena swims. - 7. red
Ben has blue. Chloe cannot have green or red, so Chloe has yellow. Ada cannot have red or blue, so Ada has green. Red is left for Dan. - 8. Gia, Finn, Eva, Hugo
Gia is 1st. If Eva were 2nd and Hugo 3rd, Finn would be last, which breaks a clue. So Finn is 2nd, Eva 3rd and Hugo 4th. - 9. Lu, Ivy, Jay, Kai
Lu > Ivy > Jay > Kai. Drawing a quick line from tall to short helps keep the comparisons straight. - 10. Clue 2
Clue 1 already gives blue to Ben, so nobody else can have blue. Clue 2 adds nothing new. Without clue 1, 3 or 4, there would be more than one answer. - 11. Tara is not in the chess club.
If Tara were in the chess club, she would also have to be in the math club. She is not, so she cannot be in the chess club. - 12. Rosa has the frisbee.
With 'Rosa has the frisbee', Pat cannot have the rope, so Pat has the soccer ball and Quinn has the rope. 'Quinn has the skipping rope' still leaves 2 answers, and repeating the first clue adds nothing. - 13. Cy
Test each person. If Ana did it, Ben and Cy both tell the truth (2 truths). If Ben did it, Ana and Cy tell the truth (2). If Cy did it, only Ben tells the truth (1). Only Cy fits. - 14. 47
Numbers from 31 to 69 with digits that add to 11: 38, 47, 56, 65. Odd ones: 47 and 65. In 65, the tens digit is bigger, so the answer is 47. - 15. Ed, Cal, Ama, Bo, Di
Di is 5th. Ed comes before Cal and Cal before Ama, and Bo comes right after Ama. That fills places 1 to 4 in only one way. - 16. Wes: poem book · Xia: dictionary · Yuri: atlas · Zoe: comic
Zoe has the comic. The dictionary cannot belong to Wes or Yuri, so it is Xia's. Yuri cannot have the poem book, so Yuri has the atlas. Wes has the poem book. - 17. Example: 'Ella is in seat 1 and Gwen is in seat 3.' Then Femi must be in seat 2.
Many clues work. Test yours: list every arrangement that fits both clues. If only one is left, your clue works.