Hints:

1. Start simple: Try experimenting with small total sums like 4, 5, 6, and see what combinations of numbers give the largest product.

2. Use only 2s and 3s: Any number 4 or greater can be broken down into a sum involving only 2’s and 3’s. For example:

   1. Replace 4 with 2 + 2

   2. Replace 5 with 2 + 3

3. Compare strategies:

   1. Is it better to use two 3’s (which sum to 6) or three 2’s (which also sum to 6)?

      • 3×3=9

      • 2×2×2=8

      • So: two 3’s are better!

4. General idea: Try to break your total sum into as many 3’s as possible, and use a 2 only when necessary.

Solution and Exploration:

Let’s start by experimenting with smaller numbers before tackling 16. Working through small cases helps us spot patterns and strategies.

Sample Breakdowns:

• 1 = 1 → Product = 1

• 2 = 2 → Product = 2

• 3 = 3 → Product = 3

• 4 = 2 + 2 → Product = 4

• 5 = 2 + 3 → Product = 2 × 3 = 6

• 6 = 3 + 3 → Product = 3 × 3 = 9

• 7 = 2 + 2 + 3 → Product = 2 × 2 × 3 = 12

• 8 = 2 + 3 + 3 → Product = 2 × 3 × 3 = 18

• 9 = 3 + 3 + 3 → Product = 3 × 3 × 3 = 27

• 10 = 2 + 2 + 3 + 3 → Product = 2 × 2 × 3 × 3 = 36

Observations:

1. Numbers greater than 4 should be broken into smaller parts—especially 2’s and 3’s.

2. Avoid using 1 unless absolutely necessary—it lowers the product.

3. Replace 4 with 2 + 2, since it doesn’t change the product but fits our pattern better.

4. Always replace 2 + 2 + 2 (product = 8) with 3 + 3 (product = 9)—a better outcome.

General Strategy:

1. If the number is even, start by expressing it as a sum of 2’s.

2. If it’s odd, start with a 3, then break the rest into 2’s.

3. Every time you have three 2’s (2 + 2 + 2), replace them with two 3’s.

Applying the Strategy:

• 16 = 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2
   → Grouped as 2 + 2 + (3 + 3) + (3 + 3)
   → Product = 2 × 2 × 3 × 3 × 3 × 3 = 324

Larger Numbers (Efficient Form):

• 20
   → Best breakdown: 2 + 3 × 6
   → Product: 2 × 3⁶ = 2 × 729 = 1458

• 50
   → Best breakdown: 2 + 3 × 16
   → Product: 2 × 3¹⁶

• 100
   → Best breakdown: 4 × 3³²
    → (e.g., 2 + 2 + 3 × 32)
   → Product: 4 × 3³²

This approach gives you the maximum product for a given sum using whole numbers—by leveraging the optimal use of 2’s and 3’s.