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arithmetic – Extra bits the merrier


The pc division at Puzzled State College devised an inefficient
bit-based
(zeros and ones) quantity system, n',
for his or her ever extra bit-happy machines.

Listed here are n' = n for 0 ≤ n ≤ 42,
skipping to X' = X the place 100000000' = X+1,
to be solved for X' and X.

    n' = n             n' = n               n' = n                n' = n

    0'   0         10000'   12  b5     100000'   24  b6     1000000'   36  b7
    1'   1  b1     10001'   13         100001'   25         1000001'   37
   10'   2  b2     10010'   14         100010'   26         1000010'   38
   11'   3         10011'   15         100011'   27         1000011'   39
  100'   4  b3     10100'   16         100100'   28         1000100'   40
  101'   5         10101'   17         100101'   29         1000101'   41
 1000'   6  b4     11000'   18         101000'   30        10000000'   42  b8
 1001'   7         11001'   19         101001'   31                  .
 1010'   8         11010'   20         101010'   32                  .
 1011'   9         11011'   21         101011'   33                  .
 1100'   10        11100'   22         101100'   34               X'   X
 1101'   11        11101'   23         101101'   35       100000000'   X+1  b9
                                                                     .
                                                                     .
  bit numbers  b1  b2  b3  b4  b5  b6  b7  b8  b9                    .
  bit values    1   2   4   6  12  24  36  42  X+1

Every n' = n is exclusive.
Solely 1000' = 6 fits n' = 6, for instance,
despite the fact that one might think about
110' = 100'+10' = 4+2 = 6 as properly.
But 110' doesn’t exist on this system.

What are the values ofX' and X?

What’s the formulaic consistency of this quantity system?

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