Problems

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Found: 1969

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 a1,a2,...,an, and the inductive hypothesis is true for all smaller sets. Then if we leave aside the person a1, everyone else a2,a3,...,an has the same color of eyes and if we leave aside an, then all a1,a2,a3,...,an1 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, n2, being a natural number, is less than or equal to n. Therefore n2n=n(n1)0. Hence, 0n1. Therefore n=1.

Find the mistake in the sequence of equalities: 1=(1)22=((1)2)12=112=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 2n1 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 2n1 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×(2n1)=2n regions.

Let’s "prove" that the number 1 is a multiple of 3. We will use the symbol to denote "congruent modulo 3". Thus, what we need to prove is that 10 modulo 3. Let’s see: 14 modulo 3 means that 2124 modulo 3, thus 216 modulo 3, however 16 gives the remainder 1 after division by 3, thus we get 21 modulo 3, next 2111 modulo 3, and thus 10 modulo 3. Which means that 1 is divisible by 3.

Recall that (n+1)2=n2+2n+1 and after expansion we get (n+1)2(2n+1)=n2. Subtract n(2n+1) from both sides (n+1)2(2n+1)n(2n+1)=n2n(2n+1) and rewrite it as (n+1)2(n+1)(2n+1)=n2n(2n+1).
Now we add (2n+1)24 to both sides: (n+1)2(n+1)(2n+1)+(2n+1)24=n2n(2n+1)+(2n+1)24.
Factor both sides into square: ((n+1)2n+12)2=(n2n+12)2.
Now take the square root: (n+1)2n+12=n2n+12.
Add 2n+12 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.
image
Can you find a mistake? Certainly the triangles have different area, so we cannot obtain one from the other one by reshaping.

This problem is often called "The infinite chocolate bar". Depicted below is a way to get one more piece of chocolate from the 5×6 chocolate bar. Do you see where is it wrong?
image

Consider the following "proof" that any triangle is equilateral: Given a triangle ABC, we first prove that AB=AC. First let’s draw the bisector of the angle A. Now draw the perpendicular bisector of segment BC, denote by D the middle of BC and by O the intersection of these lines. See the diagram
image
Draw the lines OR perpendicular to AB and OQ perpendicular to AC. Draw lines OB and OC. Then the triangles, RAO and QAO are equal, since we have equal angles ORA=OQA=90°, and RAO=QAO, and the common side AO. On the other hand the triangles ROB and QOC are also equal since the angles BRO=CQO=90°, the hypotenuses BO=OC the legs RO=OQ. Thus, AR=AQ, RB=QC, and AB=AR+RB=AQ+QC=AC. Q.E.D.

As a corollary, one can show that all the triangles are equilateral, by showing that AB=BC in the same way.