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28 February, 22:20

Given a set of weights {1,3,9,27}, show that you can construct any number between 1 and 40. In other words, using the set above and the addition and subtraction operations, construct any integer between 1 and 40 without re-using elements. For example, 4 = 1+1+1+1 is not acceptable.

For example,

8 = 9 - 1

10 = 1 + 9

+5
Answers (1)
  1. 28 February, 23:42
    0
    Yes

    Explanation:

    One way to prove it is by trying out all the possible combinations and see if the resulting set of numbers contains all the numbers from 1 to 40. You could code a program to do it omitting unnecessary calculations like those involving - 27.

    Proof that it is possible:

    1 = 1

    2 = 3-1

    3 = 3

    4 = 3+1

    5 = 9-3-1

    6 = 9-3

    7 = 9-3+1

    8 = 9-1

    9 = 9

    10 = 9+1

    11 = 9+3-1

    12 = 9+3

    13 = 9+3+1

    14 = 27-9-3-1

    15 = 27-9-3

    16 = 27-9-3+1

    17 = 27-9-1

    18 = 27-9

    19 = 27-9+1

    20 = 27-9+3-1

    21 = 27-9+3

    22 = 27-9+3+1

    23 = 27-3-1

    24 = 27-3

    25 = 27-3+1

    26 = 27-1

    27 = 27

    28 = 27+1

    29 = 27+3-1

    30 = 27+3

    31 = 27+3+1

    32 = 27+9-3-1

    33 = 27+9-3

    34 = 27+9-3+1

    35 = 27+9-1

    36 = 27+9

    37 = 27+9+1

    38 = 27+9+3-1

    39 = 27+9+3

    40 = 27+9+3+1
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