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Number Curiosities

Third Powers: The Cubes

To find the third power of a number, multiply the number by its square. For example, the third power of 3 is 3 x 9 = 27. Third powers of integers are also called cubes because n layers of n-squared unit blocks form a cube [n by n by n]. Twenty-seven one-inch cubes can be arranged into a cube 3” on a side. We say its volume is 27 cubic inches.

Looking at any one of the six faces of such a cube we see it has an area of 9 and thus a total surface area of 6 x 9 or 54. An n by n by n cube will have a surface area of 6 times n-squared.

Poser 1. What size cube has a surface area in square inches equal to one-half its volume in cubic inches?

Final Digits. Curiously, except for numbers ending in 3 or 7, the final digits of a number and its cube agree. Six-cubed is 216 and 15-cubed is 3375. Numbers ending in the three-digit groups 125, 375, 625, or 875 could be perfect cubes.

Poser 2. Find a number whose third power ends in 875.

Consecutive Integers. There is a relation among the first, second, and third powers of consecutive integers. The reader will recall from March, that the sum of 1 through n is called a triangular number – one such as 10 which is 1 + 2 + 3 + 4.

Poser 3. Find the sum of the third powers of 1 through 4. How does it compare with the triangular 10? See if the relation holds for 1 through 5.

Another connection between second and third powers is found via the odd numbers.

Poser 4. State a rule for finding square numbers (and which ones) from consecutive odd numbers.

Poser 5. Partition the odd set 1, 3, 5, 7, 9, 11, 13, … in this way: {1}, {3, 5}, {7, 9, 11}, and so on, starting where you left off and adding an additional odd to the next group. Describe how these partitions find third powers and which ones. Verify it by finding the set of odds that sum to 3375, the third power of 15.


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