Example: What happens when we start adding up odd numbers? 1=1, 1+3=4

Step 1: Observe the recursive construction from the diagrams.
Each higher-dimensional figure is built by taking two copies of the previous figure and connecting corresponding dots with edges.

The number of vertices $V_n$ in the $n$-dimensional figure satisfies:
$V_0 = 1$
$V_n = 2 \cdot V_{n-1} \quad for \quad n \geq 1$

Step 2: Verify with given diagrams.
For $n=1$: $V_1 = 2 \cdot V_0 = 2 \cdot 1 = 2$
For $n=2$: $V_2 = 2 \cdot V_1 = 2 \cdot 2 = 4$
For $n=3$: $V_3 = 2 \cdot V_2 = 2 \cdot 4 = 8$
For $n=4$: $V_4 = 2 \cdot V_3 = 2 \cdot 8 = 16$

Step 3: Solve the recurrence relation.
Unrolling the recursion:
$V_n = 2 \cdot V_{n-1} = 2^2 \cdot V_{n-2} = \cdots = 2^n \cdot V_0 = 2^n \cdot 1 = 2^n$

Step 4: This matches the powers of 2 and the number of binary sequences of length $n$ (each dot labels a sequence of $0$s and $1$s).

For the 5-dimensional figure, the number of vertices is $V_5 = 2^5$.
$2^5 = 32$

32