The dragon locked six dwarves in the cave and said, "I have seven caps of the seven colors of the rainbow. Tomorrow morning I will blindfold you and put a cap on each of you, and hide one cap. Then I’ll take off the blindfolds, and you can see the caps on the heads of others, but not your own and I won’t let you talk any more. After that, everyone will secretly tell me the color of the hidden cap. If at least three of you guess right, I’ll let you all go. If less than three guess correctly, I’ll eat you all for lunch." How can dwarves agree in advance to act in order to be saved?
Do there exist two numbers such that their sum, quotient and product would be all equal to each other?
It is easy to construct one equilateral triangle using three identical matches. Is it possible to construct four equilateral triangles by adding just three more matches identical to the original ones?
Winnie the Pooh has five friends, each of whom has pots of honey in their house: Tigger has \(1\) pot, Piglet has \(2\), Owl has \(3\), Eeyore has \(4\), and Rabbit has \(5\). Winnie the Pooh comes to visit each friend in turn, eats one pot of honey and takes the other pots with him. He came into the last house carrying \(10\) pots of honey. Whose house could Pooh have visited last?
Find the largest possible number of bishops that can be placed on the \(8 \times 8\) chessboard so that no two bishops threaten each other.
Does there exist a power of \(3\) that ends in \(0001\)?
There are \(24\) children in the class and some of them are friends with each other. The following rules apply:
If someone (say Alice) is a friend with someone else (say Bob), then the second student (Bob) is also a friend with the first (Alice).
If Alice is friend with Bob and Bob is friend with Claire, then Alice is also friend with Claire.
Find a misconception in the following statement: under the above conditions Alice is friend with herself.
Theorem: All people have the same eye color.
"Proof" by induction: This is clearly true for one person.
Now, assume we have a finite set of people, denote them as \(a_1,\, a_2,\, ...,\,a_n\), and the inductive hypothesis is true for all smaller sets. Then if we leave aside the person \(a_1\), everyone else \(a_2,\, a_3,\,...,\,a_n\) has the same color of eyes and if we leave aside \(a_n\), then all \(a_1,\, a_2,\,a_3,...,\,a_{n-1}\) also have the same color of eyes. Thus any \(n\) people have the same color of eyes.
Find a mistake in this "proof".
Let’s prove that \(1\) is the largest natural number.
Let \(n\) be the largest natural number. Then, \(n^2\), being a natural number, is less than or equal to \(n\). Therefore \(n^2-n=n(n-1)\leq 0\). Hence, \(0\leq n\leq 1\). Therefore \(n=1\).
Theorem: If we mark \(n\) points on a circle and connect each point to every other point by a straight line, the lines divide the interior of the circle is into is \(2n-1\) regions.
"Proof": First, let’s have a look at the smallest natural numbers.
When \(n=1\) there is one region (the whole disc).
When \(n=2\) there are two regions (two half-discs).
When \(n=3\) there are \(4\) regions (three lune-like regions and one triangle in the middle).
When \(n=4\) there are \(8\) regions, and if you’re still not convinced then try \(n=5\) and you’ll find \(16\) regions if you count carefully.
Our proof in general will be by induction on \(n\). Assuming the theorem is true for \(n\) points, consider a circle with \(n+1\) points on it. Connecting \(n\) of them together in pairs produces \(2n-1\) regions in the disc, and then connecting the remaining point to all the others will divide the previous regions into two parts, thereby giving us \(2\times (2n-1)=2n\) regions.