In the last post, I had written a bit about how
the last digit of increasing exponents loops in
a pattern. But that was not the most interesting
observation. One of my friends at school, had
shown that if you take the loop length of a digit,
and arrange all digits' loop length in a table, then
the loop lengths themselves cycled in a pattern.
I found this fascinating. Surely there was some
explanation for this phenomenon? I did a bit more
research, and couldn't find many satisfactory results
related to this.
I then noticed that the loop lengths,
which cycled in a 1-4-4-2-1__1-4-4-2-1 pattern,
were of loop length 5. Did this just happen to be a
coincidence, or was it because I was using base 10?
Of course, with the base 10 argument, one might
wonder why the loop lengths don't recur in five
blocks of 2?
So, the next obvious line of attack towards this
enigma was to test it out for other, numerous bases.
I started out easy, with Base 2, or Binary.
Binary, having only two digits, 0 and 1, was no
challenge. Few of us can think fluently in another
number system, so I will try linking each with more
familiar ground, the usual Base 10 system. In the
case of binary, one must consider the last digit.
If it is 1, then the equivalent decimal (Base 10)
number is odd. If the last digit is 0, then the equivalent
decimal value is even. This has a neat (and very basic)
explanation, which I suggest all my readers to try
and figure out. Back to binary - An odd number's exponents
will always be odd, and an even number's exponents
will always be even. Therefore, 0-ending numbers result in
0, and 1-ending numbers result in 1. This gives each
possibility a loop-length of 1. So binary's loop-lengths
recur in a pattern of 1_1. So far so good.
With binary tackled, I hope to pursue tertiary/ternary's
properties now. That I hope, will let us see a little deeper
into the mystery.
the last digit of increasing exponents loops in
a pattern. But that was not the most interesting
observation. One of my friends at school, had
shown that if you take the loop length of a digit,
and arrange all digits' loop length in a table, then
the loop lengths themselves cycled in a pattern.
I found this fascinating. Surely there was some
explanation for this phenomenon? I did a bit more
research, and couldn't find many satisfactory results
related to this.
I then noticed that the loop lengths,
which cycled in a 1-4-4-2-1__1-4-4-2-1 pattern,
were of loop length 5. Did this just happen to be a
coincidence, or was it because I was using base 10?
Of course, with the base 10 argument, one might
wonder why the loop lengths don't recur in five
blocks of 2?
So, the next obvious line of attack towards this
enigma was to test it out for other, numerous bases.
I started out easy, with Base 2, or Binary.
Binary, having only two digits, 0 and 1, was no
challenge. Few of us can think fluently in another
number system, so I will try linking each with more
familiar ground, the usual Base 10 system. In the
case of binary, one must consider the last digit.
If it is 1, then the equivalent decimal (Base 10)
number is odd. If the last digit is 0, then the equivalent
decimal value is even. This has a neat (and very basic)
explanation, which I suggest all my readers to try
and figure out. Back to binary - An odd number's exponents
will always be odd, and an even number's exponents
will always be even. Therefore, 0-ending numbers result in
0, and 1-ending numbers result in 1. This gives each
possibility a loop-length of 1. So binary's loop-lengths
recur in a pattern of 1_1. So far so good.
With binary tackled, I hope to pursue tertiary/ternary's
properties now. That I hope, will let us see a little deeper
into the mystery.
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