National Aeronautics and Space Administration

B. Examine the procedure below.

In this example, the beginning number falls into the numerical 'black hole' 1. Repeat the same steps with a starting value of 97. What number results?

Does starting with 81 yield the same type of numerical 'black hole' as starting with 97? Let us investigate:

81 --> 8² +1² or 65, which becomes 6² + 5² or 61. Following this through, you'll see the following pattern emerge:

81 --> 65 --> 61 --> 37 --> 58 --> 89 --> 145 --> 42 --> 20 --> 4 --> 16 --> 37 --> 58...

Notice that this sequence has begun to repeat and, in fact, falls into a cyclic numerical 'black hole'. The sequence eventually becomes periodic.

C. Now try 204. Does this create a new numerical 'black hole'? Justify your answer.

D. Can you describe without computation what happens if you start the procedure with 420?

The heavy object representing the black hole will distort the latex surface (representing spacetime) and cause the small objects on the surface to be pulled in toward it... but not if you are too far away. A heavier ball bearing, however, would affect beads further out in the latex sheet...just as a more massive black hole creates a larger distortion in spacetime, thus affecting objects further away.