Circles and lines are drawn on the plane. They divide the plane into non-intersecting regions, see the picture below.
Show that it is possible to colour the regions with two colours in such a way that no two regions sharing some length of border are the same colour.
Consider a number consisting of \(3^n\) digits, all ones, such as 111 or 111111111 for example. Show that such a number with \(3^n\) digits is divisible by \(3^n\).
Numbers \(1,2,\dots,n\) are written on a whiteboard. In one go Louise is allowed to wipe out any two numbers \(a\) and \(b\), and write their sum \(a+b\) instead. Louise enjoys erasing the numbers, and continues the procedure until only one number is left on the whiteboard. What number is it? What if instead of \(a+b\) she writes \(a+b-1\)?
Prove that
(a) \[1^2 + 2^2 + 3^2 + \dots + n^2 = \frac{1}{6} n (n+1)(2n+1)\]
(b) \[1^2 + 3^2 + 5^2 + \dots + (2n-1)^2 = \frac{1}{3} n (2n-1)(2n+1)\].
Is “I see what I eat” the same thing as “I eat what I see”?
To make it not so confusing let’s change the wording to make it more “mathematical”
“I see what I eat”=“If I eat it then I see it”
“I eat what I see”= “If I see it then I eat it”
Was the March Hare right? Is “I like what I get” the same thing as “I get what I like”?
Do you remember the example from the previous maths circle?
“Take any two non-equal numbers \(a\) and \(b\), then we can write; \(a^2 - 2ab + b^2 = b^2 - 2ab + a^2\).
Using the formula \((x-y)^2 = x^2 - 2xy + y^2\), we complete the squares and rewrite the equality as \((a-b)^2 = (b-a)^2\).
As we take a square root from the both sides of the equality, we get \(a-b = b-a\). Finally, adding to both sides \(a+b\) we get \(a-b + (a+b) = b-a + (a+ b)\). It simplifies to \(2a = 2b\), or \(a=b\). Therefore, All NON-EQUAL NUMBERS ARE EQUAL! (This is gibberish, isn’t it?)”
Do you remember what the mistake was? In fact we have mixed up two things. It is indeed true “if \(x=y\), then \(x^2 = y^2\)”. But is not always true “if \(x^2 = y^2\), then \(x=y\).” For example, consider \(2^2 = (-2)^2\), but \(2 \neq (-2)!\) Therefore, from \((a-b)^2 = (b-a)^2\) we cannot conclude \(a-b = b-a\).
Was the Dormouse right? Is “I breathe when I sleep” the same thing as “I sleep when I breathe”?
Mary Ann yawns every time it rains. In fact Mary Ann is yawning right now. Is it raining at the moment?
The March Hare, the Hatter, and the Dormouse were accused of stealing some tarts. At the trial The March Hare said it was the Hatter who stole the tarts. The Dormouse and the Hatter made statements too, but nobody remembered what they said, and all the records were destroyed by Alice’s tears. As the trial proceeded, it became clear that the tarts were stolen by a person, who in fact was the only one who gave a truthful statement. Who stole the tarts?