## Sum of Consecutive Cubes (Visual Proof)

The sum of the first n cubes is the square of the nth triangular number:
13 + 23 + 33 + 43 + 53 + . . . + n3 = (1 + 2 + 3 + 4 + 5 + . . . + n)2.

## Inverse Powers of Phi

Summation of Alternating Inverse Powers of Phi…

## Fibonacci Right Triangle

The sum of the squares of consecutive Fibonacci numbers is another Fibonacci number.

## Visual Proof (sum of cubes)

The sum of the sequence of the first n cubes equals [n(n+1)/2]² as shown below:
1³+2³+3³+…+n³ = (1+2+3+…+n)² = [n(n+1)/2]²

## “Stubborn” Number 33

It is conjectured that n is a sum of 3 cubes if n is a number that is not congruent to 4 or 5 mod 9. The number 33 enters this category, but for 64 years no solutions emerged — that is, whether the equation 33 = x³ + y³ + z³ has an integer solution. Continue reading ““Stubborn” Number 33″

## Sangaku: Semicircle inscribed in a right triangle

Find the radius r of the semicircle inscribed in the right triangle below:

show solution

## Math Ambigram

(by Peter Rowlett)
Solve this equation for x. Then rotate 180° and solve for x again.
The equation works either ways as 𝑥, 1, 8 and the unusual 5 have rotational symmetry. The same is true of +, =, and the horizontal line in a fraction.

## “Magic” Factorials

There are many fun facts regarding the factorials. For instance:

• 0! = 1 by convention. As weird as it may sound, this is a fact that we must remember.
• The number of zeroes at the end of n! is roughly n/4.
• 70! is the smallest factorial larger than a googol.
• The sum of the reciprocals of all factorials is e.
• Factorials can be extended to fractions, negative numbers and complex numbers by the Gamma function.

It is possible to “peel” each layer off of a factorial and create a different factorial, as shown in the neat number pattern below. A prime pattern can be found when adding and subtracting factorials. Alternating adding and subtracting factorials, as shown in the picture, yields primes numbers until you get to 9! Continue reading ““Magic” Factorials”

## Prime Fractions

Did you know? You can write the number 1 as a sum of 48 different fractions, where every numerator is 1 and every denominator is a product of exactly two primes.

This problem is related to the Egyptian fractions.

## Hydrostatic Solution Of Particular Trinominial Equations

A. Demanet devised an interesting method of solution of trinomial equations which depends on the use of communicating vessels of convenient forms.
To solve an equation of the third degree of the form:
x3 + x = c
where c is a constant, an inverted cone and a cylinder, joined together by means of a tube, are taken.
As shown below.

The radius r of the cone and its height h are in the ratio:
r/h = √3/√π
while the base of the cylinder is taken as 1 cm2