1.6 Summary & Closing Puzzle
Summary
Key Points
- A number multiplied by itself is a square number; squares of natural numbers are perfect squares (1, 4, 9, 16, \dots), written n^2.
- Perfect squares end only in 0, 1, 4, 5, 6 or 9, and finish with an even number of zeros. (Ending in 2,3,7,8, or an odd run of zeros, rules a square out.)
- The sum of the first n odd numbers equals n^2. To square a number ending in 5, compute a(a+1) then append 25.
- The square root undoes squaring: \sqrt{n^2} = \pm n. Every perfect square has a positive and a negative integer root; we usually take the positive one, \sqrt{\ }.
- A number is a perfect square exactly when its prime factors split into two identical groups.
- A number multiplied by itself three times is a cube, written n^3 (1, 8, 27, \dots). A number has exactly three factors exactly when it is the square of a prime.
- A number is a perfect cube when its prime factors split into three identical groups; the cube root is written \sqrt[3]{\ }, e.g. \sqrt[3]{27} = 3.
A Closing Puzzle: Square Pairs!
Try This Look at this row: \quad 6 \quad 3 \quad 1 \quad 8 \quad 17. Each neighbouring pair adds to a perfect square:
6+3=9,\quad 3+1=4,\quad 1+8=9,\quad 8+17=25.
Now your turn: arrange the numbers 1 to 15 (each used once) in a row so that every neighbouring pair adds to a perfect square. Is there more than one way? For an extra challenge, try arranging 1 to 23 in a row under the same rule.