The original question includes a diagram: A 3 by 3 grid of squares, each containing a seated figure representing one person.
Nine people are sitting in the squares of a 3 by 3 grid, one in each square. Two people are called neighbours if they are sitting in squares that share a side. (People in diagonally adjacent squares, which only have a point in common, are not called neighbours.)
Each of the nine people in the grid is either a truth-teller who always tells the truth, or a liar who always lies.
Every person in the grid says: 'My neighbours are all liars'.
Given only this information, what are the smallest number and the largest number of people who could be telling the truth?
- Asmallest 1, largest 4
- Bsmallest 2, largest 4
- Csmallest 2, largest 5
- Dsmallest 3, largest 4
- Esmallest 3, largest 5
- Fsmallest 4, largest 4
- Gsmallest 4, largest 5
- Hsmallest 5, largest 5
Show the answer and worked solution
answer · E
- Asmallest 1, largest 4
- Bsmallest 2, largest 4
- Csmallest 2, largest 5
- Dsmallest 3, largest 4
- Esmallest 3, largest 5
- Fsmallest 4, largest 4
- Gsmallest 4, largest 5
- Hsmallest 5, largest 5
Turn the words into two rules. A truth-teller has no truth-telling neighbour, so no two truth-tellers are adjacent; a liar's sentence must be false, so every liar has at least one truth-telling neighbour. For the largest, take the four corners and the centre: none of these five is adjacent to another, and each of the four edge squares touches at least one of them, so 5 works, and 6 or more would force two truth-tellers to be adjacent. For the smallest, try three, say the top-left corner, the middle of the right-hand column and the bottom-left corner — no two are adjacent and every remaining square touches one of them. Two cannot work: two non-adjacent squares cover at most seven squares between them, and checking the possible pairs always leaves some liar with no truth-telling neighbour, which would make that liar's statement true.