Proof: \( (1+2+3+…+n)^2 =1^3 + 2^3 + 3^3 + … + n^3 \).
Explains this Image:
\[ (1+2+3+…+8)^2 = 1^3+2^3+3^3+…+8^3 \]
\[ S(\text{square}) = (1+2+3+…+8)\times (1+2+3+…+8) \]
\[ S(\text{square}) = (1+2+3+…+8)^2 \]
Also, \[ S(\text{square}) = \text{SUM of small squares} \]
\[= 1\times 1^2 + 2\times(2^2) + 3\times(3^2) + 4\times(4^2)+…+8\times(8^2) \]
\[ = 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 \]
In there, pink square and white square are compensate for each other.
See more: 3D geometry proof posted by Hyrodium’s Graphical MathLand &TwoCubes.
Image: Carre de la somme des 8 premiers entiers on Wikipedia.
What Is an Example of a Counterintuitive Mathematical Result?
The Hydra Game will always lead to a result that you probably wouldn’t expect.
This is anaptly named blog. Check out all the answers to this question on Quora for more counterintuitive results!
The Dragon Curve
This is a fractal resembles a dragon and is also known as the Jurassic Park Fractal.
Starting from a base segment, replace each segment by 2 segments with a right angle and with a rotation of 45° alternatively to the right and to the left. Continue this with the newly generated segments.
consider a large rectangle R which is tiled with small rectangles of various shapes(but with sides vertical or horizontal), each of which possess the following property P: ” one of the sides of the rectangle has integer length.”
prove the rectangle R has itself has the property P.
Each dot is only moving in a straight line, but is created by balls moving in circles through 3 dimensional space.
This is the magic hexagon. The numbers add up to 38 along each straight line. It can be called the magic hexagon rather than a magic hexagon because there are no other hexagons numbered 1,2,…,n with this property, no matter how many layers the arrangement has (except for the one which is just one hexagon with 1 written in it, but that’s hardly magical…).
(Source: Mathematical Gems I by Ross Honsberger)
Two curves cut all circles at right angles: straight line and a tractrix.
I didn’t know what a tractrix was, so I Googled it! Tractrix definition from Wikipedia: Tractrix (from the Latin verb trahere ”pull, drag”; plural: tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a tractor (pulling) point that moves at a right angle to the initial line between the object and the puller at an infinitesimal speed.
Here’s a gif from Wikipedia showing a tractrix being created from dragging a pole:
— | Calculus professor (via mathprofessorquotes) |
timb:
How a hole is drilled to be made square, the red shape in the center would be the cutting tool.
it shares the same principle as a Reuleaux triangle but with one rounded corner so that the cut square does not have rounded edges; the cutting tool follows the path of the rounded edge that is tangent to the sides of the outer square. because a tangent line is perpendicular to the radius, as the cutting tool follows the path of the rounded edge it turns precisely 90° to create a sharp edged perfect square.
— | David Hilbert, upon hearing one of his students stopped attending his lectures, in favor of poetry. (via curiosamathematica) |