Calculate how many of an item it takes to make a larger item.

Question

Example: I collect gems and they are graded by size (#1 to #12). The only way for me to get a larger sized gem is to trade 2 smaller sized gems. So I need 2 #1's in order to get a #2 gem. Then I need 2 #2's to get a # 3 gem and so on. Question: How many #1 gems will I need to get up to a #12 gem? What is the formula Is it 2•1=2•2=4•2=8•2=16 and I do this 12 times? Wouldn't that be 4,496? Thanks for your time and knowledge.

Answer 1

For a #12 gem, you need $2$ #11 gems, thus $2×2=4$ #10 gems, thus $2×2×2=8$ #9 gems and so on – $2^{12-k}$ $\#k$ gems are needed. Hence $2^{12-1}=2048$ #1 gems.

This exponential growth explains why 2048 games can be really long and challenging, because you are only given (the equivalent of) #1 and #2 gems and need to make (the equivalent of) a #11 gem.