Compute successive differences over levels for perfect cubes until all the differences at a level are the same. What do you notice?

Question

Homi · Accepted Answer

We will find the differences between consecutive perfect cubes, and then the differences between those differences, until we reach a level where all numbers are the same.

Step 1 — List Perfect Cubes

Let us list the first few perfect cubes. A perfect cube is a number obtained by multiplying an integer by itself three times.

$1^3 = 1$

$2^3 = 8$

$3^3 = 27$

$4^3 = 64$

$5^3 = 125$

$6^3 = 216$

$\boxed{\text{Perfect Cubes: } 1, 8, 27, 64, 125, 216, \dots}$

Step 2 — Compute Level 1 Differences

Now, we find the differences between consecutive perfect cubes. This is our first level of differences.

$8 - 1 = 7$

$27 - 8 = 19$

$64 - 27 = 37$

$125 - 64 = 61$

$216 - 125 = 91$

$\boxed{\text{Level 1 Differences: } 7, 19, 37, 61, 91, \dots}$

Step 3 — Compute Level 2 Differences

Next, we find the differences between consecutive numbers from Level 1. This is our second level of differences.

$19 - 7 = 12$

$37 - 19 = 18$

$61 - 37 = 24$

$91 - 61 = 30$

$\boxed{\text{Level 2 Differences: } 12, 18, 24, 30, \dots}$

Step 4 — Compute Level 3 Differences

Finally, we find the differences between consecutive numbers from Level 2. This is our third level of differences.

$18 - 12 = 6$

$24 - 18 = 6$

$30 - 24 = 6$

$\boxed{\text{Level 3 Differences: } 6, 6, 6, \dots}$

Diagram 1

Answer

We notice that the differences become constant at Level 3. All the differences at Level 3 are 6. This is similar to how the differences for perfect squares became constant at Level 2 (with a constant difference of 2). The level at which the differences become constant is equal to the power of the numbers (e.g., 2 for squares, 3 for cubes).

Question 30

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