Puzzle 321
A Proof That 2 = 1. Let A = B:
A = B
multiply by A on both sides:
A x A = A x B
subtract B x B from both sides:
A x A − B x B = A x B − B x B
factorise:
(A + B)(A − B) = B(A − B)
divide by (A − B) on both sides:
(A + B) = B
replace A with B:
B + B = B
factorise:
2B = B
divide by B on both sides:
2 = 1
Where lies the flaw in the logic?
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Answer
On line 4 we are dividing both sides by (A − B).
But we know that A = B, so A − B = 0.
Dividing by zero is undefined, so we are not allowed to do this.
Puzzle 323
What is the smallest positive integer which …
… when multiplied by 3 gives a number consisting of only 4's?
Puzzle Copyright © Kevin Stone
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Hint
Try working backwards.
Answer
148.
Reasoning
A number that is divisible by 3 has digits that add up to a number that is also divisible by 3 (for example, 123 is divisible by 3).
So we can look for the first number where 4 + 4 + 4 + 4 + 4 + … is divisible by 3, which is 4 + 4 + 4, and 444 ÷ 3 = 148.
Alternative Reasoning
Let's try the possible answers in order:
4 ÷ 3 = 1.3333
44 ÷ 3 = 14.6666
444 ÷ 3 = 148
Double-Checking
148 x 3 = 444.
Puzzle 324
What letter is missing from this sequence:
A · B · G · D · E · ? · E
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Hint
This is quite a common alphabet.
Answer
Z.
From Zeta, first letters of the Greek alphabet, alpha, beta, gamma, delta, epsilon, zeta, eta.
