# Brain teasers

## Brain teasers

**Jack is looking at Anne, but Anne is looking at George. Jack is married, but George is not. Is a married person looking at an unmarried person?**

A) Yes.

B) No.

C) Cannot be determined.

A) Yes.

B) No.

C) Cannot be determined.

This is from this month’s Scientific American. It’s about “dysrationalia,” which is what happens when people with nominally high IQ’s end up thinking irrationally

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## Re: Brain teasers

Arithmetic and algebraic paradoxes

Proving that 2 = 1

Here is the version offered by Augustus De Morgan: Let x = 1. Then x² = x. So x² - 1 = x -1. Dividing both sides by x -1, we conclude that x + 1 = 1; that is, since x = 1, 2 = 1.

Quine, p.5

Assume that

a = b. (1)

Multiplying both sides by a,

a² = ab. (2)

Subtracting b² from both sides,

a² - b² = ab - b² . (3)

Factorizing both sides,

(a + b)(a - b) = b(a - b). (4)

Dividing both sides by (a - b),

a + b = b. (5)

If now we take a = b = 1, we conclude that 2 = 1. Or we can subtract b from both sides and conclude that a, which can be taken as any number, must be equal to zero. Or we can substitute b for a and conclude that any number is double itself. Our result can thus be interpreted in a number of ways, all equally ridiculous.

Northrop, p. 85

The paradox arises from a disguised breach of the arithmetical prohibition on division by zero, occurring at (5): since a = b, dividing both sides by (a - b) is dividing by zero, which renders the equation meaningless. As Northrop goes on to show, the same trick can be used to prove, e.g., that any two unequal numbers are equal, or that all positive whole numbers are equal.

Proving that 2 = 1

Here is the version offered by Augustus De Morgan: Let x = 1. Then x² = x. So x² - 1 = x -1. Dividing both sides by x -1, we conclude that x + 1 = 1; that is, since x = 1, 2 = 1.

Quine, p.5

Assume that

a = b. (1)

Multiplying both sides by a,

a² = ab. (2)

Subtracting b² from both sides,

a² - b² = ab - b² . (3)

Factorizing both sides,

(a + b)(a - b) = b(a - b). (4)

Dividing both sides by (a - b),

a + b = b. (5)

If now we take a = b = 1, we conclude that 2 = 1. Or we can subtract b from both sides and conclude that a, which can be taken as any number, must be equal to zero. Or we can substitute b for a and conclude that any number is double itself. Our result can thus be interpreted in a number of ways, all equally ridiculous.

Northrop, p. 85

The paradox arises from a disguised breach of the arithmetical prohibition on division by zero, occurring at (5): since a = b, dividing both sides by (a - b) is dividing by zero, which renders the equation meaningless. As Northrop goes on to show, the same trick can be used to prove, e.g., that any two unequal numbers are equal, or that all positive whole numbers are equal.

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