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Fifteen Jumbled (Posted on 2006-12-29) Difficulty: 3 of 5
The numbers 1 through 15 have been re-ordered according to a certain logic. Deduce the logic and fill in the blanks.

15, 7, 11, 3, ___, ___, ___, 1, 14, 6, ___, ___, 12, 4, 8

See The Solution Submitted by Jer    
Rating: 3.0000 (2 votes)

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Solution Puzzle Solution Comment 7 of 7 |
(In reply to Answer by K Sengupta)

At the outset, the first term of the sequence is 15, which is 
represented as 1111 in base two (binary) system. Reversing the digits,
we also obtain 1111, which when converted to its decimal representation,
becomes 15.
The third term is 11, represented as 1011 in base 2. Reversing its 
digits, we obtain 1101, which when converted back to base 10 is 13.
Accordingly, we can conjecture that the given numbers are obtained 
by writing the decimal numbers 15, 14, 13, ...., 2, 1 in turn, in
their respective base two representation in 4 digits, reversing the
digits, and then converting each of the said rteversed numbers to
obtain the terms of the given sequence.
Let 
S(n) = nth term of the given sequence, and:
B(p) = Binary representation of the decimal number p, in four digits.
R(p) = Number constituted by reversing the digits of B(p).
D(p) = Decimal (base 10) representation of the binary number R(p).
Accordingly, we constitute the following table:
p        B(p)      R(p)       D(p)
15       1111      1111        15
14       1110      0111         7
13       1101      1011        11
12       1100      0011         3 
11       1011      1101        13
10       1010      0101         5
9        1001      1001         9
8        1000      0001         1 
7        0111      1110        14   
6        0110      0110         6 
5        0101      1010        10  
4        0100      0010         2 
3        0011      1100        12 
2        0010      0100         4 
1        0001      1000         8
From the above table, it is verified that S(p) = D(16-p) for all the 
given values of p.
Consequently, the missing 5th term, 6th term and the 7th term are 
respectively 13, 5 and 9; while the missing 11th term and the 12th
term are respectively 10 and 2.
 

  Posted by K Sengupta on 2008-07-02 15:47:04
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