Solution

Great, but how do I deal with a large number of nuts?

Of course, you can always systematically derive all combinations for any number of nuts using either of the above approaches. The key here is to find the pattern, and the way to do that is to start as simple as possible, starting with 1 nut and working up until a pattern emerges -- from experience, you may need to work through 5 to 7 nuts. Regardless of which approach you take, the total number of combinations is the same, but I'll use the second approach and explain later in the advanced section.

Here are the different combinations for different numbers of nuts:

• 1 nut: 1 way
  • 1
• 2 nuts: 2 ways
  • 1,1
  • 2
• 3 nuts: 3 ways
  • 1,1,1
  • 1,2
  • 2,1
• 4 nuts: 5 ways
  • 1,1,1,1
  • 1,1,2
  • 1,2,1
  • 2,1,1
  • 2,2
• 5 nuts: 8 ways
  • 1,1,1,1,1
  • 1,1,1,2
  • 1,1,2,1
  • 1,2,1,1
  • 1,2,2
  • 2,1,1,1
  • 2,1,2
  • 2,2,1
• 6 nuts: 13 ways
  • 1,1,1,1,1,1
  • 1,1,1,1,2
  • 1,1,1,2,1
  • 1,1,2,1,1
  • 1,1,2,2
  • 1,2,1,1,1
  • 1,2,1,2
  • 1,2,2,1
  • 2,1,1,1,1
  • 2,1,1,2
  • 2,1,2,1
  • 2,2,1,1
  • 2,2
• 7 nuts: 21 ways
  • 1,1,1,1,1,1,1
  • 1,1,1,1,1,2
  • 1,1,1,1,2,1
  • 1,1,1,2,1,1
  • 1,1,1,2,2
  • 1,1,2,1,1,1
  • 1,1,2,1,2
  • 1,1,2,2,1
  • 1,2,1,1,1,1
  • 1,2,1,1,2
  • 1,2,1,2,1
  • 1,2,2,1,1
  • 2,1,2,2
  • 2,1,1,1,1,1
  • 2,1,1,1,2
  • 2,1,1,2,1
  • 2,1,2,1,1
  • 2,1,2,2
  • 2,2,1,1,1
  • 2,2,1,2
  • 2,2,2,1

Continue to solution explained