42: The Answer To Life, The Universe and Everything
Tags: programming, java, jdk
Published: 2020-08-04 19:39:21

---
title: "42: The Answer To Life, The Universe and Everything"
published: true
description: 
tags: programming,java,jdk
---
The story started with a [Twitter Post](https://twitter.com/lukaseder/status/1254133314962759688) about 
the JDK method `bitCount` which is available for `Long` and `Integer` types.

If you take a look into the time line, there was a reply https://twitter.com/ScottSelikoff/status/1254185742760280064 by @ScottSelikoff (funny comment of course) who stated `42` as the answer to life, the universe and everything to the magic method which has been followed by [Lukas Eder](https://twitter.com/lukaseder/status/1254283473327710208?s=20):

> For what percentage of longs would that be the correct result?

This thread has inspired me to this article.

So let us summarize that question: 
> For what percentage of long values is `bitCount()` == 42?

The first question which arises: What does `bitCount` do? Let us take a look into the java doc of the
function (Integer variant):
> Returns the number of one-bits in the two's complement binary 
> representation of the specified int value. This function is sometimes
> referred to as the population count.

So in the end `bitCount` counts the number (as the name implies) of `1`s which are in a given value.
 
Here are some exemplary values for the type `Integer`:

* `Integer i = 0b1100_0000_0000_0000_0000_0000_0000_0000` `bitCount=2`
* `Integer i = 0b0000_0000_0000_0000_0000_0000_0000_0011` `bitCount=2`
* `Integer i = 0b0000_0000_0000_0000_0000_0000_0000_1111` `bitCount=4`
* `Integer i = 0b1111_1111_1111_0000_0000_0000_0000_0000` `bitCount=12`
* `Integer i = 0b1010_1010_1010_1010_1010_1010_0000_0000` `bitCount=12`
* `Integer i = 0b1111_1111_1111_1111_1111_1111_1111_1111` `bitCount=32`

A very naive way of trying to solve that question would be to write code like this:
```java
BigDecimal pow = BigDecimal.valueOf(2L).pow(64);

Map<Integer, Long> collect = LongStream.range(Long.MIN_VALUE, Long.MAX_VALUE)
  .boxed()
  .collect(groupingBy(Long::bitCount, counting()));

collect.forEach((k, v) -> System.out.printf("k = %4s v=%15s %7.5f\n", k, v, BigDecimal.valueOf(v)
  .divide(pow)
  .multiply(BigDecimal.valueOf(100L))));
```
The first line will create `2^64` which is equal to 100% and the 
`BigDecimal.valueOf(v).divide(pow).multiply(BigDecimal.valueOf(100L))` will just calculate the percentage
of the value (number of values for appropriate bit count) of the resulting map which contains the number.

Ok now we can try that.... Really? Never. This code will run a very very long time...

> A long time ago in a galaxy far, far away...

That might be a little long to wait for the result. Ok let's think a bit about the code. Ah! Yes of course I got it. 
We should use `parallel()` for the stream to get it faster.

```java
BigDecimal pow = BigDecimal.valueOf(2L).pow(64);

Map<Integer, Long> collect = LongStream.range(Long.MIN_VALUE, Long.MAX_VALUE)
  .boxed()
  .parallel()
  .collect(groupingBy(Long::bitCount, counting()));

collect.forEach((k, v) -> System.out.printf("k = %4s v=%15s %7.5f\n", k, v, BigDecimal.valueOf(v)
  .divide(pow)
  .multiply(BigDecimal.valueOf(100L))));
```
That version will be faster than the previous one and will take... Sorry but I simply don't know cause I have not tested it ;-).

Can we make a simpler and faster solution to get a result in the end? Ok we change the `Long` into `Integer`? Now the code looks like this. As you see I already added the `parallel()`
in the code to get faster also you see I'm using `BigDecimal.valueOf(2L).pow(32)` (2^32) instead of 2^64 
based on the usage of `Integer`: 

```java
BigDecimal pow = BigDecimal.valueOf(2L).pow(32);

Map<Integer, Long> collect = IntStream.range(Integer.MIN_VALUE, Integer.MAX_VALUE)
  .boxed()
  .parallel()
  .collect(groupingBy(Integer::bitCount, counting()));

collect.forEach((k, v) -> System.out.printf("k = %4s v=%15s %7.3f\n", k, v, BigDecimal.valueOf(v)
  .divide(pow)
  .multiply(BigDecimal.valueOf(100L))));
```
So this code will run in ca. 15 seconds (Hexa Core CPU's) with the following result:

```
k =    0 v=              1   0.000
k =    1 v=             32   0.000
k =    2 v=            496   0.000
k =    3 v=           4960   0.000
k =    4 v=          35960   0.001
k =    5 v=         201376   0.005
k =    6 v=         906192   0.021
k =    7 v=        3365856   0.078
k =    8 v=       10518300   0.245
k =    9 v=       28048800   0.653
k =   10 v=       64512240   1.502
k =   11 v=      129024480   3.004
k =   12 v=      225792840   5.257
k =   13 v=      347373600   8.088
k =   14 v=      471435600  10.976
k =   15 v=      565722720  13.172
k =   16 v=      601080390  13.995
k =   17 v=      565722720  13.172
k =   18 v=      471435600  10.976
k =   19 v=      347373600   8.088
k =   20 v=      225792840   5.257
k =   21 v=      129024480   3.004
k =   22 v=       64512240   1.502
k =   23 v=       28048800   0.653
k =   24 v=       10518300   0.245
k =   25 v=        3365856   0.078
k =   26 v=         906192   0.021
k =   27 v=         201376   0.005
k =   28 v=          35960   0.001
k =   29 v=           4960   0.000
k =   30 v=            496   0.000
k =   31 v=             31   0.000
k =   32 v=              1   0.000
```

The first column `k=` is the `bitCount` so the first line means:
 
We have a single value `v=1` where the `bitCount==0` and `0.000` percent. 
 
Let use take a value on the line for `k=16` and `v=601,080,390` and `13,995` percent of the values of Integer. So it means for `bitCount=16` we have `601,080,390` values which are `13,995` percent of all integer values.

One interesting observation which can be made here is that the number of values is rising up to a maximum value
at `k=16` (`bitCount=16`) which you might not have expected. Another thing is that the number of values is going 
down to the higher number with `bitCount=32` etc.  

Based on the runtime of this example you could roughly estimate the runtime for the variant with Long:
`15 s * 2^32 = 64.424.509.440 seconds / 86400 seconds / day =  745.654 days / 365 days / year` which results in ca. 2,000 years. So having a machine which has 1000 times more CPUs it could be dropped down to about 2 years (theoretically) or maybe you could use more power by using GPU's on AWS cloud but in the end no realizable solution.

So we need to reconsider if there is a faster or easier solution to answer the question? Yes there is one.

The answer can be found by using some [mathematics](https://en.wikipedia.org/wiki/Permutation#Permutations_with_repetition).

Let us take a known example from the above result output:

We have 32 bits (Integer). Now we have 16 bits which should be `0` and 16 bits which should be `1`. By expressing that
like this:
{% katex %}
 \frac{32!}{16!\cdot16!} = 601080390
{% endkatex %}
You can calculate that via [WolframAlpha](https://www.wolframalpha.com/input/?i=32%21%2F%2816%2116%21%29) and the result
is: `601080390` which is exact the number in the above example. Let us check some other values:

{% katex %}
 \frac{32!}{9!\cdot23!} = 28048800
{% endkatex %}

The resulting value will be two times in the table one for `k=9` (`bitCount=9`) and for `k=23` (`bitCount=23`). The `bitCount=9` means 9 `1`s (and 23 `0`s) are in the integer value and `bitCount=23` means 23 `1`s (and 9 `0`) are in the integer values.

So based on the mathematics we could really answer the question via:

{% katex %}
 \frac{64!}{42!\cdot22!} = 80,347,448,443,237,920
{% endkatex %}

So this means we have 80,347,448,443,237,920 numbers where the `bitCount=42` and this means
{% katex %}
 \frac{80,347,448,443,237,920}{2^{64}}\cdot 100 = 0,435 %
{% endkatex %}

So `0,435` percent of all long values having a `bitCount=42`.