Cake Number

What do you understand by the term `Cake Number`? Let's decode it in this article.

The term "cake number" describes a concept of discrete geometry and combinatorics−related mathematical idea. It is built on the concept of the Lazy caterer's sequence.

What is the Lazy Caterer's Sequence?

The maximum number of pieces a disk (cake or pizza) can be sliced into using a specific number of straight slices is known as the Lazy caterer's sequence. Although it mentions a disk, we will consider a cake in our example. One straight cut can divide a cake into two pieces, two straight cuts can divide a cake into four pieces, and three straight cuts can divide a cake into seven pieces. The following formula can be used to determine the cake number for n cuts:

C(n) = (n^2 + n + 2) / 2

You might still be confused. Let me take an example. Here’s a way to cut a cake into 7 pieces with 3 straight cuts:

• Slice the cake in half, evenly, and horizontally.

• Make a vertical cut through the center of the two sections after stacking them on top of one another.

• Then, from the top of the cake to the bottom, create a diagonal cut that intersects the preceding one.

The cake will be cut into two half−circles, two quarter−circles, and three triangular pieces after these three cuts, forming a total of seven pieces.

There may be various ways to cut a cake into 7 pieces with 3 straight cuts in addition to this method of cutting. Yet, no matter how the cuts are made, the greatest number of pieces that may be obtained with three straight slices is always seven. This is because the cake number formula C(3) = (32 + 3 + 2) / 2 = 7.

This concept is extended to 3−dimension which gives birth to the cake number.

What is the Cake Number?

For a given number of planes, the cake number refers to the maximum number of regions a 3−D cube can be partitioned.

The formula to calculate the cake number for a given number of planes is:

Cake Number = (n3 + 5*n + 6) / 6

Computer science, operations research, and social choice theory are a few of the disciplines that use cake numbers.

Approach

Here is the step−by−step approach to calculating the cake number.

• Take the number of planes as user input.

• Store the value in integer variable n.

• Use the formula Cake Number = (n3 + 5*n + 6) / 6 to calculate the maximum number of regions the 3−d cube can be divided into using the given number of planes.

• Return the value and print it to the console.

C++ Code Implementation to Calculate Cake Number

Now, we have covered a lot of theory, let’s get straight to code.

Example

#include <iostream>

using namespace std;

int cake_number(int n) {
    return (n*n*n + 5*n + 6) / 6;
}

int main() {
    int n = 4;
    
    int result = cake_number(n);
    cout << "The cake number for " << n << " planes is " << result << endl;
    return 0;
}

Output

The cake number for 4 planes is 15

Time Complexity: O(1)

Space Complexity: O(1)

Conclusion

In this article, we covered what is cake number and how to calculate a cake number for the given number of straight cuts as input. Hope you are able to grasp the concept well and found the article useful.