Given that 1
· 2
· 3
… · 30
= 1 , and all of the numbers are integers,
Is it possible for 1
+ 2
+ 3
+ ⋯+ 30
= 0 ?
We observe that all the numbers must be any integers, so we can deduce that all numbers must be either +1 or -1, since for example, having a number like +2 would require a -1/2 as a pair to create the +1 in the answer for the first equation. In order for the second equation to be possible, we must have the same number of +1s and -1s so that together they give an answer of 0. This means we have in total fifteen pairs of +1s and -1s. If we try to approach the first equation, since we know that the product of a pair (+1 × −1) gives -1 with the parity rules, the product of these fourteen pairs must give +1. Wait, there’s the 15 th pair left! So +1 x -1 gives an answer of -1, which isn’t the answer we are seeking from the first equation. We can thus conclude that these two equations are impossible having used rules of parity.
Now let’s look at another scenario below:
One day, you and nine other friends are walking down the street and an UFO secretly lands beside you. All 10 of you are captured by these aliens and now they put you to a test to investigate whether humans are intelligent life forms. If you succeed, you will be safely released back to Earth.
They will put you into a row of 10 in descending heights and facing the front, so that the tallest person at the back can see everyone in front, and the second tallest person can see the 8 people in front of him etc. Then, either black or white hats are put on everyone in a completely random fashion, and you can only see the colour of the hat of the people in front of you. In order for all of you to be released, at least 9 people have to guess correctly of the colour of their own hat. The order of the guesses has to be
from the tallest person to the shortest. Unfortunately, if you say anything other than “black” or “white”, or try to do anything suspicious by changing the tone of your voice or other actions, you will be fried instantly by their laser gun. Everyone can hear the guesses of all the previous people. Now, before you are put to the test, you all are allowed 10 minutes to discuss and form a plan. Can you overcome this test? 1 Firstly, we observe that there are only two possible colours of hats, which means it is possible we can use ideas of parity to solve this, either a white hat, or a non-white hat (black). The method is as follows: the tallest person establishes the parity of the number of white hats. So he could say “black” to suggest that he sees an odd number of white hats, or say “white” if he sees an even number. He has a 50% of guessing the colour of his own hat correctly, but this doesn’t matter – as long as the other 9 gets it right. Using the above picture as an example, the tallest person will say “white” as he sees 6 people with white hats. This tells the second tallest person that he sees an even number of white hats, but the second person can only see an odd number of 5 white hats in front of her. The second tallest person
1 https://www.youtube.com/watch?v=N5vJSNXPEwA&vl=en – TED talk by Alex Gendler
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