How Many Numbers Does Moritz Underline Exactly Twice?

In a unique mathematical challenge, the whole numbers from 1 to 2016 are written on a blackboard. Moritz underlines specific multiples in various colors: all multiples of two in red, all multiples of three in blue, and all multiples of four in green. The question is, how many numbers does Moritz underline exactly twice?

Identifying the Multiples

To solve this problem, we begin by identifying the numbers that are multiples of two, three, and four. Here, we leverage the properties of divisibility and integer division operations.

Multiples of 2 (Red): All even numbers from 1 to 2016. Multiples of 3 (Blue): All numbers divisible by 3 from 1 to 2016. Multiples of 4 (Green): All numbers divisible by 4 from 1 to 2016.

The counts of these multiples can be calculated as follows:

Count of multiples of 2 (Red): mathleftlfloor frac{2016}{2} rightrfloor 1008/math Count of multiples of 3 (Blue): mathleftlfloor frac{2016}{3} rightrfloor 672/math Count of multiples of 4 (Green): mathleftlfloor frac{2016}{4} rightrfloor 504/math

Identifying Overlaps

Next, we need to identify numbers that are multiples of combinations of these three numbers but are not multiples of all three.

Multiples of 6 (2 and 3): mathleftlfloor frac{2016}{6} rightrfloor 336/math Multiples of 12 (3 and 4): mathleftlfloor frac{2016}{12} rightrfloor 168/math Multiples of 4 (2 and 4): mathleftlfloor frac{2016}{4} rightrfloor 504/math (already counted) Multiples of 12 (2, 3, and 4): mathleftlfloor frac{2016}{12} rightrfloor 168/math

Applying the Principle of Inclusion-Exclusion

Using the principle of inclusion-exclusion, we can determine the numbers that are underlined exactly twice. This principle helps in finding the count of elements that belong to at least two of the sets without overcounting those that belong to all three sets.

Multiples of 2 and 3: 336 Multiples of 2 and 4: 504 (already counted) Multiples of 3 and 4: 168

Subtracting the overlap (multiples of 12):

Total numbers considered: 336 504 168 1008 Subtracting the overlap: 1008 - 168 840 Already counted: 504

Thus, the number of numbers that Moritz underlines exactly twice is:

text{Exactly twice} 336 - 168 504 - 168 504

A Programmatic Solution

For a more automated solution, we can write a program to count the numbers that are underlined exactly twice. The Scala code can be written as:

```scala(1 to 2016).map(x > List(x % 2  0, x % 3  0, x % 4  0)).filter(_.count(_  true)  2).sizeres0: Int  504```

This program iterates through the numbers from 1 to 2016, creates a list of booleans for each number indicating whether it is a multiple of 2, 3, or 4, filters the list to keep only those numbers where exactly two booleans are true, and counts the results.

To conclude, the total number of numbers that Moritz underlines exactly twice is 504.