¡Juguemos Kenken!

(This column originally appeared in the July 2009 issue of La Voz, in Spanish.)

Welcome back to the puzzle column. This time we have two new kenken puzzles for you. The first one is a 4×4 kenken that uses only addition, and the second a 5×5 kenken made up entirely of multiplication.

For your convenience, we repeat the rules of kenken:

Similar to Sudoku, your task is to fill in the 4×4 grid using only the numbers 1, 2, 3 and 4 (and the 5×5 grid using only the numbers 1,2,3,5 and 5) in such a way that:
1. Each number appears only once per row.
2. Each number appears only once per column.
3. The numbers in each cage, when combined with the operation given, yield the target number.

Since kenken means “wisdom squared” in Japanese, we know you’ll get a little smarter by successfully completing these puzzles!


The second kenken puzzle is entirely multiplication, which means that a good strategy to use is factorization. Look at the top middle cage, for example. We have to find three numbers whose product is 20. Multiplication has the amazing property that the order of the factors doesn’t matter, so listing the factors in non-decreasing order, there are four ways: 20 = 1×1×20 = 1×2×10 = 1×4×5 = 2×2×5. Of course we immediately eliminate the first two ways, since we are only allowed to fill in the grid with the numbers 1, 2, 3, 4 and 5.

Using factorization on the bottom right cage is even more effective: 25 = 1×1×25 = 1×5×5. The first factorization uses 25, which is too big. The second factorization has a repeated factor, which in combination with rules 1 and 2 above, allow us to conclude the unique way to fill in the cage.

Factorization leads to some amazing patterns. Consider the cage in the middle of the puzzle that prompts us to express 12 as the product of two numbers. The solutions are 12 = 1×12 = 2×6 = 3×4. The numbers that appear, 1, 2, 3, 4, 6 and 12, are called the divisors of 12. There is an amazing way to arrange the divisors of 12, shown below. Moreover, by counting 1 and the number 12 itself, 12 has 6 divisors.

This figure is called the divisor lattice of 12. Notice that moving up and to the right (northeast) is equivalent to multiplying by 3. Likewise, moving up and to the left is equivalent to multiplying by 2. From a tilted perspective, the divisors of 12 form a 2×3 grid.

Problem C: Draw the divisor lattices for the numbers from 1 to 20. What do 2, 3, 5 and 7 have in common? Can you draw the divisor lattice for 30? Is 1 a prime number? What other patterns do you notice?

Problem D: What is the smallest number with exactly 2009 divisors? (count 1 and the number as divisors).

If you enjoyed these puzzles, I’d love to hear from you. Please send in your solutions to any of the problems A, B, C or D as a comment.