On an island each person only ever tells the truth or only ever lies. Ana says Ben and she are the same kind; Ben says Cara is truthful; Cara says Ana lies. How many are liars?
This is the tier's one puzzle with no arithmetic at all — only what each person's words force about the others. The pivot is a small, delightful lemma: a claim of the form "so-and-so and I are the SAME kind" gives the other person away no matter who says it. If a truth-teller says it, it is true and they match; if a liar says it, it is false and they differ from a liar — either way the other one is a truth-teller. "Different kinds" is the mirror.
Why this sheet works
Once one person is pinned that way, every other statement is a plain claim by someone whose kind is now known, so it settles its subject outright. The whole puzzle unrolls in a straight line — anchor, then pin, then pin — with no guessing and no case-splitting. The answer asked for is a single number, how many of the three are liars, so there is nothing to argue about once the reasoning is done.
How this one works
One question from a sheet like this one, worked a step at a time — with a sentence saying what just happened and why.
Work out how many of the islanders are liars, from what each of them says.
1On an island, every person is either a truth-teller, who only says true things, or a liar, who only says false things. Maya says, "Leo and I are of different kinds." Leo says, "Nell is a liar." Nell says, "Maya always tells the truth."How many of the three are liars?Why
ArithmeticWho is lying
Name: Date:
Work out how many of the islanders are liars, from what each of them says.
2On an island, every person is either a truth-teller, who only says true things, or a liar, who only says false things. Elsa says, "Dev and I are of different kinds." Dev says, "Faye always tells the truth." Faye says, "Elsa always tells the truth."How many of the three are liars?Why
ArithmeticWho is lying
Name: Date:
Work out how many of the islanders are liars, from what each of them says.
3On an island, every person is either a truth-teller, who only says true things, or a liar, who only says false things. Hana says, "Gus and I are the same kind." Gus says, "Iris always tells the truth." Iris says, "Hana is a liar."How many of the three are liars?Why
ArithmeticWho is lying
Name: Date:
Work out how many of the islanders are liars, from what each of them says.
4On an island, every person is either a truth-teller, who only says true things, or a liar, who only says false things. Hana says, "Gus and I are of different kinds." Gus says, "Iris is a liar." Iris says, "Hana is a liar."How many of the three are liars?Why
Who is lyingAnswer key
1. Start with Maya's statement, because a "same kind / different kind" claim gives the OTHER person away whoever makes it. If Maya tells the truth, then Maya and Leo really are of different kinds, so Leo is a liar. If Maya is lying, they are really the same kind, and since Maya would be a liar, Leo is a liar too. Either way, Leo is a liar. Now Leo says "Nell is a liar". We know Leo is a liar, so that is a lie — which makes Nell a truth-teller. Then Nell says "Maya always tells the truth". As a truth-teller, that is true, so Maya is a truth-teller. So Leo is a liar, Maya is a truth-teller and Nell is a truth-teller — 1 of them is a liar.2. Start with Elsa's statement, because a "same kind / different kind" claim gives the OTHER person away whoever makes it. If Elsa tells the truth, then Elsa and Dev really are of different kinds, so Dev is a liar. If Elsa is lying, they are really the same kind, and since Elsa would be a liar, Dev is a liar too. Either way, Dev is a liar. Now Dev says "Faye always tells the truth". We know Dev is a liar, so that is a lie — which makes Faye a liar. Then Faye says "Elsa always tells the truth". As a liar, that is false, so Elsa is a liar. So Dev is a liar, Elsa is a liar and Faye is a liar — 3 of them are liars.3. Start with Hana's statement, because a "same kind / different kind" claim gives the OTHER person away whoever makes it. If Hana tells the truth, then Hana and Gus really are the same kind, so Gus is a truth-teller. If Hana is lying, then they are really of different kinds, and since Hana would be a liar, Gus is a truth-teller once more. Either way, Gus is a truth-teller. Now Gus says "Iris always tells the truth". We know Gus is a truth-teller, so that is true — which makes Iris a truth-teller. Then Iris says "Hana is a liar". As a truth-teller, that is true, so Hana is a liar. So Gus is a truth-teller, Hana is a liar and Iris is a truth-teller — 1 of them is a liar.4. Start with Hana's statement, because a "same kind / different kind" claim gives the OTHER person away whoever makes it. If Hana tells the truth, then Hana and Gus really are of different kinds, so Gus is a liar. If Hana is lying, they are really the same kind, and since Hana would be a liar, Gus is a liar too. Either way, Gus is a liar. Now Gus says "Iris is a liar". We know Gus is a liar, so that is a lie — which makes Iris a truth-teller. Then Iris says "Hana is a liar". As a truth-teller, that is true, so Hana is a liar. So Gus is a liar, Hana is a liar and Iris is a truth-teller — 2 of them are liars.
