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An American family


Mr. and Mrs. Smith and their two children form a typical American family. If exactly two of the following statements are true, what is the first name of the father, mother, son and daughter:

  • George and Dorothy are blood relatives.
  • Howard is older than George.
  • Virginia is younger than Howard.
  • Virginia is older than Dorothy.

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Assume first is true.

So either George and Dorothy are brother and sister or Father and daughter or Mother and son.

First case:

Brother and sister

Second statement is true Howard would be the father.

Third could be true or false. (Mother could be older or young than Father)

Forth would be true. (Mother would be older than daughter)

So are can’t be brother and sister. (Three true statements)

Second Case:

Father and Daughter

Second statement is false.

Third would be false too.

Third statement is true.

(Two true and two false)

No need to carry on:

Father is George

Mather is Virginia

Son is Howard

and Daughter is Dorothy.

Making first and forth statements true and second and third statements false.

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If [1] is true, then G and D could be: (a) son-daughter, (b) son-mother, © father-daughter.

If [2] is true, then H and G could be: father-son

If [3] is true, then V and H could be (a) daughter-father, (b) mother-father, © daughter-son.

If [4] is true, then V and D could be: mother-daughter.

If [2] & [4] were the exactly two true statements, then [1] also becomes true, which violates the condition given in OP.

[1] & [3] have multiple choices, and if they were the exactly two true statements, then: CASE-I: [1a] & [3b], or CASE-II: [1b] & [3a], could be true.

In both the above cases, [2] also becomes true, which is violation of the condition given in OP.

It is therefore clear that only one of the statement out of [1} & [3] and [2] & [4] must be True.

If [1] Is True, then [3] has to be false.

But only if [1c] was true, only then [3] could be false; and only if [3c] was true, only then [1] could be false.

So, let us assume [1c] as true, i.e. G – father, D – daughter; then [3] & [2] become false and [4] could be true; which satisfies the condition in OP. Therefore, George is father, Dorothy is daughter, Virginia is mother, and Howard is son.

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There is one other option that no-one seems to have considered ..... an incestuous marriage?

Statements 1 and 3 can be correct, with George as dad, Dorothy as mum, and the two children Howard and Virginia.

Or does that go bryond the 'typical American family' guideline?

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Howard is Papa. He's a man, and older than George.

George is son. He's male, and younger that his father.

Dorothy is George's blood relative. Either mother or sister.

Since Virginia is older than Dorothy, that makes her mother.

Which leaves Dorothy as George's sister.

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Following the conventional method:

Dad=Howard (H) or George (G)

Mom=Virgina (V) or Dorothy (D)

Option 1: Dad=H,Mom=V leads to statements 1,2 & 4 being T and 3 could be either T or F. Does not work.

Option 2: Dad=G, Mom=V leads to statements 1,4 being T and 2,3 being F. We can stop here since we have the solution.

But just to be exhaustive -

Option 3: Dad=H, Mom=D leads to 3 T statements

Option 4: Dad=G, Mom=D leads to potentially 4 F statements unless "incestuous" relationships are considered, in which case there could be 2 T and 2 F statements. We can dismiss this as "ambiguous".

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