Inspiring and Creative Resources & Tutorials for Science-Curious People
How to easily solve a nested radical equation,,,
You can adapt it to any repeating basic math operator in the infinitely nested radicals. The above nested radicals can be equal to whatever you want except 1, this is due to the particular multiplicative identity properties of the digit 1 (for instance, √1 = 1 and 1² = 1).
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.
The sum of the squares of consecutive Fibonacci numbers is another Fibonacci number.
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
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″
Find the radius r of the semicircle inscribed in the right triangle below:
show solution
(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.
There are many fun facts regarding the factorials. For instance:
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”
