Maximizing Products with 16
Look at ways to break up 16 into a sum of numbers that you can then multiply to get as big a product as possible. Writing 16 = 10 + 6 is a start, but 16 = 6 + 5 + 5 is better. Can you do better?
16 = 10 + 6 and 10 x 6 = 60
16 = 6 + 5 + 5 and 6 x 5 x 5 = 150
THE CHALLENGE
What is the biggest product you can make by breaking 16 into a sum of numbers?
EXPLORATION
How does your strategy change if you replace 16 with 20, 50, or 100?
Notes
THE CHALLENGE & EXPLORATION
Let’s start by looking at some examples that replace 16 with smaller numbers. Getting this experience is almost always a good way to start, and it usually doesn’t take long.
1 = 1 => 1
2 = 2 => 2
3 = 3 => 3
4 = 2 + 2 or just 4 => 4
5 = 2 + 3 => 2 x 3 = 6
6 = 3 + 3 => 3 x 3 = 9
7 = 2 + 2 + 3 => 2 x 2 x 3 = 12
8 = 2 + 3 + 3 => 2 x 3 x 3 = 18
9 = 3 + 3 + 3 => 3 x 3 x 3 = 27
10 = 2 + 2 + 3 + 3 => 2 x 2 x 3 x 3 = 36
As you look at these examples, a pattern emerges for breaking down larger numbers.
- Any number larger than 4 should be replaced with smaller numbers!
- Never use 1 unless you have no choice.
- Use 2 + 2 instead of 4. They give the same result, but it is easier to see how to improve things when looking at 2’s rather than 4’s.
- Always replace 2 + 2 + 2, which gives 2 x 2 x 2 = 8, with 3 + 3, which gives 3 x 3 = 9.
Using these rules, a general strategy emerges for attacking any number.
- If the number is even, write it as a sum of 2’s. If the number is odd, write it as 3 plus a sum of 2’s.
- Replace any group of three 2’s with a group of two 3’s.
With all this in place, we are ready to solve the problem.
16 = 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 2 + 2 + 3 + 3 + 3 + 3, which gives a product of 2 x 2 x 3 x 3 x 3 x 3= 324.
As for the other numbers, we have:
20 = 10 x 2 = 2 + 3 x (3 x 2) = 2 + 3 x (2 x 3) = 2 + 6 x 3, which gives 2 x 3^6 = 2 x 729 = 1458.
50 = 25 x 2 = 2 + 8 x (3 x 2) = 2 + 8 x (2 x 3) = 2 + 16 x 3, which gives 2 x 3^16.
100 = 50 x 2 = 2 x 2 x 16 x (3 x 2) = 2 + 2 + 16 x (2 x 3) = 2 x 2 + 32 x 3, which gives 4 x 3^32.