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.
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.
