### 1. Squares with Twin Halves

36,363,636,364^{2} = 1,322,314,049,613,223,140,496

The first 11 digits of the square are the same as the last 11 digits. This is the smallest possible solution for this pattern.

Here’s another one:

63,636,363,637^{2} = 4,049,586,776,940,495,867,769.

Furthermore, adding these two numbers results to:

36,363,636,364 + 63,636,636,637 = 100,000,000,001.

### 2. Consecutive Squares with a Cube Sum

The sum of consecutive squares of 22 to 68 is equal to 103,823 or 47^{3}:

47^{3} = 22^{2} + 23^{2} + 24^{2} … + 66^{2} + 67^{2} + 68^{2}

### 3. The Number 1,729

Srinivasa Ramanujan

According to Srinivasa Ramanujan, “[1,729] is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.” It can be expressed as:

1,729 = 1^{3} + 12^{3} = 9^{3} + 10^{3}

In addition, if you add the number of the beast (666) to 1,729, the total is equal to the sum of the first prime number (2) and the squares of the next consecutive nine primes.

1,729 + 666 = 2 + 3^{2} + 5^{2} +7^{2} + 9^{2} + 11^{2} + 13^{2} + 17^{2} + 19^{2} + 23^{2} + 29^{2 }= 2,395

### 4. Only Single Possible Solutions

• 88^{2} = 7744

It is the only solution of its kind.

• 1^{2} + 2^{2} + 3^{2} + … + 23^{2} + 24^{2} = 70^{2}

This is also the only solution for the problem of the sum of consecutive squares beginning with 1 having a total of a square number.

### 5. Münchhausen Numbers

A Münchhausen number is equal to the sum of its digits raised to each digit’s own power. There are only two of these numbers beside the obvious 0 and 1. See A046253.

3,435 = 3^{3} + 4^{4} + 3^{3} + 5^{5}

438,579,088 = 4^{4} + 3^{3} + 8^{8} + 5^{5} + 7^{7} + 9^{9} + 0^{0} + 8^{8} + 8^{8}

I agree with Sue. RE-posted on twitter @trefology

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The beauty of Math. Well written!

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These are beyond my brain capacity for numbers but make interesting reading. Thanks for my early morning challenge.

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