The nine clues in today’s puzzle are all perfect squares. They are all you need to find all the factors that can turn this puzzle into a multiplication table . . . but with the rows and columns not in the typical order:

Print the puzzles or type the solution in this excel file: 10-factors-1087-1094

1089 is also a perfect square.

The previous perfect square was 32² = (33 – 1)² = 33² + 1 – 2(33) = 1024

The next perfect square will be 34² = (33 + 1)² = 33² + 1 + 2(33) = 1156

Here’s a little more about the number 1089:

- 1089 is a composite number.
- Prime factorization: 1089 = 3 × 3 × 11 × 11, which can be written 1089 = 3²× 11²
- The exponents in the prime factorization are 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1) = 3 × 3 = 9. Therefore 1089 has exactly 9 factors.
- Factors of 1089: 1, 3, 9, 11, 33, 99, 121, 363, 1089
- Factor pairs: 1089 = 1 × 1089, 3 × 363, 9 × 121, 11 × 99, or 33 × 33
- 1089 is a perfect square. √1089 = 33

1, 9, 121, and 1089 are all perfect square factors of 1089.

1089 can be 3 × 3 perfect squares arranged on an 11 × 11 perfect square grid:

1089 can also be 11 × 11 perfect squares arranged on a 3 × 3 perfect square grid:

Not only is 1089 the 33rd perfect square, but it is also the sum of the first 33 odd numbers. Note that the *nth* perfect square is also the sum of the first *n *odd numbers:

I’m not attempting to make a picture of this nine-sided shape, but 1089 is the 18th nonagonal number because 18(7(18) – 5)/ 2 = 1089,

or written another way 7(18²)/2 – 5(18)/2 = 1089.

1089 is the sum of five consecutive prime numbers:

199 + 211 + 223 + 227 + 229 = 1089

OEIS.org informs us that 9 × **1****089** = **980****1**

1089 looks rather square when it is written in several other bases:

It’s 900 in BASE 11 because 9(11²) = 1089,

441 in BASE 16 because 4(16²) + 4(16) + 1(1) = 1089,

169 in BASE 30 because 1(30²) + 6(30) + 9(1) = 1089,

144 in BASE 31 because 1(31²) + 4(31) + 4(1) = 1089,

121 in BASE 32 because 1(32²) + 2(32) + 1(1) = 1089,

100 in BASE 33 because 1(33²) = 1089