Cake Number in Java

In this section, we will learn about finding the cake number in Java. The cake number CK_n represents the maximum number of pieces a cake can be divided with the help of n planar cuts. The tridimensional version of the pancake numbers is the cake numbers.

In general,

CK_n = (n³ + 5 *n + 6) / 6 where n >= 0

Thus,

For n = 0,

CK₀ = (0³ + 5 * 0 + 6) / 6 = (6) / 6 = 6 / 6 = 1

For n = 1,

CK₁ = (1³ + 5 * 1 + 6) / 6 = (1 + 5 + 6) / 6 = 12 / 6 = 2

For n = 2,

CK₂ = (2³ + 5 * 2 + 6) / 6 = (8 + 10 + 6) / 6 = 24 / 6 = 4

For n = 3,

CK₃ = (3³ + 5 * 3 + 6) / 6 = (27 + 15 + 6) / 6 = 48 / 6 = 8

For n = 4,

CK₄ = (4³ + 5 * 4 + 6) / 6 = (64 + 20 + 6) / 6 = 90 / 6 = 15

For a = 5,

CK₅ = (5³ + 5 * 5 + 6) / 6 = (125 + 25 + 6) / 6 = 156 / 6 = 26

Implementation

Let's see how one can implement the above mathematical formula. Observe the following example.

FileName: CakeNumberExample1.java

Output:

The first 30 cake numbers are: 
1 2 4 8 15 26 42 64 93 130 176 232 299 378 470 576 697 834 988 1160 1351 1562 1794 2048 2325 2626 2952 3304 3683 4090

Using Recursion

The cake numbers can also be found using recursion. However, to find cake numbers using recursion, we must know the recursive formula of it.

CK₀ = 1, for n = 0

CK₁ = 2, for n = 1

CK_n = ^{n + 1}C₃ + n + 1, for n >= 2

Thus,

For n = 2

Implementation

Let's see how one can implement the above mathematical formula. Observe the following example.

FileName: CakeNumberExample2.java

Output:

The first 30 cake numbers are: 
1 2 4 8 15 26 42 64 93 130 176 232 299 378 470 576 697 834 988 1160 1351 1562 1794 2048 2325 2626 2952 3304 3683 4090

Using Iteration

The cake numbers can also be found using iteration. We will use loops to find the cake numbers with the help of a two-dimensional array. Observe the following program.

FileName: CakeNumberExample3.java

Output:

The first 30 cake numbers are: 
1 2 4 8 15 26 42 64 93 130 176 232 299 378 470 576 697 834 988 1160 1351 1562 1794 2048 2325 2626 2952 3304 3683 4090

Another Approach

Apart from the above-discussed approach, there is another approach to compute the cake numbers using combinations. Observe the following mathematical statement.

CK₀ = 1, for

CK₁ = 2, for

CK₂ = 4, for

CK_n = ⁿC₃ + ⁿC₂ + ⁿC₁ + ⁿC₀, for n >= 3

Thus,

For n = 3,

Implementation

Let's see the recursive as well as the iterative implementation of the above formula to find the cake numbers in the code mentioned below.

FileName: CakeNumberExample4.java

Output:

The first 30 cake numbers are: 
1 2 4 8 15 26 42 64 93 130 176 232 299 378 470 576 697 834 988 1160 1351 1562 1794 2048 2325 2626 2952 3304 3683 4090 

The first 30 cake numbers are: 
1 2 4 8 15 26 42 64 93 130 176 232 299 378 470 576 697 834 988 1160 1351 1562 1794 2048 2325 2626 2952 3304 3683 4090