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Decimal and Binary Representation (Posted on 2016-12-13) Difficulty: 3 of 5
Find a number with the following properties:
1. When expressed in decimal the number has each of the ten digits occuring exactly once.
2. When expressed in binary the number has the same number of 1s and 0s.

No leading zeros are allowed.

No Solution Yet Submitted by Brian Smith    
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Hints/Tips hint based somewhat on computer findings | Comment 1 of 3
A computer program finds that these numbers are plentiful enough that, after assuring that numbers you try are in the range where the binary representation has an even number of digits, you won't have to try very many before finding an example. (On the order of 1 in 10 numbers with even number of binary digits.)

Start with the knowledge that  1023456789 has 30 digits when expressed in binary (its hexadecimal representation is 3D00B615).  9876543210 has 34 digits when expressed as binary (its base-2 log is  33.2013590408944, a different way of determining the number of binary digits).  Fitting numbers will have 30, 32 or 34 binary digits, and, as said, many of them will have equal numbers of 1's and 0's.

Edited on December 13, 2016, 10:41 am
  Posted by Charlie on 2016-12-13 10:31:41

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