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Consider the integer powers of 7: a(n) = 7^n:
7, 49, 343, 2401, 16807,... 117649, ...

Prove that the second from right digit in all those numbers (excluding the 1st) is either 0 or 4.

Three concentric circles each have radii 8, 15 and 17.

Find the largest possible area of the equilateral triangle with one vertex on each circle.

Begin with a finite sequence of blocks in a row, each in one of 3 colors: red, blue, yellow.

Below each pair of neighboring blocks place a new block with the color rule: If the blocks are the same color use that color but if they are different use the third color.

Example:

r b y y b
 y r y r
  b b b
   b b
    b

How can the color of the last block be easily predicted from the top row?

Note: I don't know the full answer but can solve special cases.

Find the side length of the smallest equilateral triangle in which three discs of radii of 2, 3, 4 can be placed without overlap.

What is the highest prime number with no repeated digits?

List all the positive integers which are equal to the sum of the digits of their respective cubes.

How many of them are prime?

What is the 100th number in this sequence?

7261, 7264, 7265, 7274, 7278, 7302, 7303, 7310, 7317, 7334, 7337, 7336, 7363, 7357, 7366, 7406, 7413, 7425, 7450, 7452, 7462, 7495, 7500, 7563, 7568, 7623, 7644, 7691, 7698, 7723, 7740, 7751, 7803, 7808, 7831, 7844, 7855, 7863, 7894, 7905, 7925, 7935, 7979, 8001, 8169, 8264, 8273, 8310, 8327, 8482, 8524, 8571, 8622, 8661, 8679, 8785, 8839, 8925, 8948, 9077, 9085, 9119, 9164, 9223, 9242, 9347, 9431, 9494, 9537, 9610, 9730, 9783, 9808, 9862, 9980, 10004, 10053, 10254, 10358, 10476, 10535, 10605, 10677, 10731, 10815, 10870, 10891, 10912, 11000, 11080, 11130, 11177, 11201, 11242, 11346, 11379, 11407, 11464, 11507, ?

The plot of sin (x + y) = cos (x² + y²) looks like a collection of an infinite number of circles. Find the angle of intersection between the smallest two circles in this family of circles.

If p is the perimeter of a right angle triangle with integral length sides, {a,b,c}, there are exactly three solutions for p = 120.
{20,48,52}, {24,45,51}, {30,40,50}

For which value of p ≤ 1000, is the number of solutions maximized?

Source: Project Euler

Consider an ellipse fixed at the origin of the xy-plane having semi-major axis 4 and semi-minor axis 3. A congruent ellipse rolls over the first ellipse without slipping. Initially, they share a common major axis.

Find the locus of focus (the one nearer to the origin) of the rolling ellipse, and calculate the distance traveled by the focus in one revolution (around the fixed ellipse).

This number N can be expressed as the sum of 1, 2, 3, 4, 5, 6, 7, 8 distinct squares.

Find N and show the 8 possible partitions.

Rem: This fact was discovered by Crespi de Valldaura.

49 and 1681 are squares with an even number of digits, where both halves are squares, and no zeroes are used.
List all such numbers below 10^10.

Though this be madness, yet there is method in't.
(Hamlet Act 2, scene 2)

Sunday: 6176
Monday: 62816
Tuesday: 731021
Wednesday: 941336
Thursday: 851340
Friday: 661236
Saturday: ???

What number replaces "???" ?
What is " the method in't "?

List all two digit numbers that if reversed and added to the original number create a perfect square.