Integer chemistry II

Level 2

Integers, like molecules, are built from atoms. However there's only one element in the periodic table of maths, namely the number 1, and all other integers can be built from it. For example, there are several ways to build the number 6:

6 = 1 + 1 + 1 + 1 + 1 + 1 = ( 1 + 1 ) ∗ ( 1 + 1 ) + 1 + 1 = ( 1 + 1 ) ∗ ( 1 + 1 + 1 ) 6 = 1+1+1+1+1+1 \\ = (1+1)*(1+1)+1+1 \\ = (1+1)*(1+1+1)

We say the number 6 has complexity 5, because that's the cheapest (using the fewest ones) way it can be built. The allowed operations are addition, multiplication and brackets. It's not allowed to write two ones next to each other to make 11.

What's the complexity of the number 31?


The answer is 11.

Matt Mundy by Matt Mundy , 32, United Kingdom

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6 solutions

Since 31 is a prime number, we can only optimize the complexity of 31 by optimizing the nearest non-prime number i.e 30, whose prime factorization is 2 ∗ 3 ∗ 5 2*3*5 .

Each of these individual blocks can be represented only by 1s added to themselves, giving 30 a complexity of 10.

Ultimately, 31 ends up having (complexity (30))+1 = 11 \boxed{11}

5 Helpful 1 Interesting 0 Brilliant 0 Confused
Sadie Robinson
Feb 2, 2014

( 1 + 1 ) ∗ ( 1 + 1 + 1 ) ∗ ( 1 + 1 + 1 + 1 + 1 ) + 1 = 2 ∗ 3 ∗ 5 + 1 = 30 + 1 = 31 (1 + 1) * (1 + 1 + 1) * (1 + 1 + 1 + 1 + 1) + 1 = 2 * 3 * 5 + 1 = 30 + 1 = 31

In total, 2 1's * 3 1's * 5 1's + 1 = 31, means 11 1's were used. 11 \boxed {11}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

31 = (10 * 3) + 1

The complexity of 10 is 7

The complexity of 3 is 3

The complexity of 1 is 1

So, we have 7 + 3 + 1 = 11

0 Helpful 0 Interesting 0 Brilliant 0 Confused

By the rules below it is how i solved

(1+1+1+1+1)x(1+1+1)x(1+1)+1=31

(5)x(3)x(2)+1=31

sum those number and complexity of the number 31 is..

(5)+(3)+(2)+1=11

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Naimish Khara
Feb 1, 2014

it is 11 by this method 31=(1+1) (1+1+1) (1+1+1+1+1)+1 so total no. of "1" is 11 thus, it;s complexity is 11

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I got 11 by (1+1) *(1+1) *(1+1) *(1+1) *(1+1) -1 = 31

Leon Fernandes - 7 years, 4 months ago
Vikram Chugh
Jan 31, 2014

(1+1+1) * (1+1+1+1+1) * (1+1) +1

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