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Sample Brainteaser

Emperor Zorg’s Kingdom

Emperor Zorg of the alien planet Zorgoid adores his two alien-squid daughters so much that he decrees that all fellow Zorgians in his kingdom would be better off having more girl alien-squid offspring than boy alien-squid offspring.

“I hereby announce that all Zorgian couples, denizens of the planet Zorgoid, must continue to have children until they have a daughter!”

“Furthermore, all child-bearing Zorgians will stop having children as soon as they have a daughter!”

After aeons have passed, what is the expected ratio of girls to boys on planet Zorgoid, assuming that the chance of giving birth to a boy is the same as a girl at 50%?

Show Answer

This is an excellent brainteaser as it triggers most people to overthink the solution…

Since the chance of having a boy or a girl is 50%, and therefore the ratio of boys to girls is 1:1, the planet of Zorgoid will have approximately 50% girls and 50% boys after a long period of time has elapsed. 

If you suppose there are X child-bearing alien couples on Zorgoid, half of these will have just the single child under Emperor Zorg’s rules – if they have a girl, then they must stop there. 

Half of the remaining half (X/4), will have two children, a boy and a girl. Half of the remaining quarter (N/8) will have 3 children, 2 boys and 1 girl, etc. 

This algebraic series is as follows: X + X/2 + X/4 + X/8 + X/16, etc. 

The sum of this series is always 2X – there will always be 1 girl for every boy. 

Essentially, then, the very fact that the probability of having a boy is the same as a girl is the answer to the riddle. Whilst this probability remains 50%, Emperor Zorg’s decree will have no long-term effect on the number of boys and girls.

It would take some time for the balance to be achieved, as there may be scenarios where families continue to have boys for a long period of time before they have a girl. 

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