Problems

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?

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.

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 $n^2-n=n(n-1)\le 0$. Hence, $0\le n\le 1$. Therefore $n=1$.

Find the mistake in the sequence of equalities: $-1=(-1{)}^{\frac{2}{2}}=((-1{)}^2{)}^{\frac{1}{2}}=1^{\frac{1}{2}}=1$.

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.

Let’s "prove" that the number $1$ is a multiple of $3$. We will use the symbol $\equiv$ to denote "congruent modulo $3$". Thus, what we need to prove is that $1\equiv 0$ modulo $3$. Let’s see: $1\equiv 4$ modulo $3$ means that $2^1\equiv 2^4$ modulo $3$, thus $2\equiv 16$ modulo $3$, however $16$ gives the remainder $1$ after division by $3$, thus we get $2\equiv 1$ modulo $3$, next $2-1\equiv 1-1$ modulo $3$, and thus $1\equiv 0$ modulo $3$. Which means that $1$ is divisible by $3$.

Recall that $(n+1{)}^2=n^2+2n+1$ and after expansion we get $(n+1{)}^2-(2n+1)=n^2$. Subtract $n(2n+1)$ from both sides $(n+1{)}^2-(2n+1)-n(2n+1)=n^2-n(2n+1)$ and rewrite it as $(n+1{)}^2-(n+1)(2n+1)=n^2-n(2n+1)$.
Now we add $\frac{(2n+1{)}^2}{4}$ to both sides: $(n+1{)}^2-(n+1)(2n+1)+\frac{(2n+1{)}^2}{4}=n^2-n(2n+1)+\frac{(2n+1{)}^2}{4}$.
Factor both sides into square: $((n+1)-\frac{2n+1}{2}{)}^2=(n-\frac{2n+1}{2}{)}^2$.
Now take the square root: $(n+1)-\frac{2n+1}{2}=n-\frac{2n+1}{2}$.
Add $\frac{2n+1}{2}$ to both sides and we get $n+1=n$ which is equivalent to $1=0$.

Look at the following diagram, depicting how to get an extra cell by reshaping triangle.

Can you find a mistake? Certainly the triangles have different area, so we cannot obtain one from the other one by reshaping.