For a rainy day
- Start with the numbers 1, 2, 3, and 4.
- What do I do with them?
- Add them up, that’s what you do with them.
- That is easy. I get 9.
- Try again.
- Did I say 9? It was a slip of tongue, I meant 10.
- Now you have five numbers, 1, 2, 3, 4, and 10.
- I feel rich!
- Pick any four of them and add them.
- Why on earth would I do that?
- To get a bigger number, you silly.
- What is the aim?
- It may stop the rain.
- Seriously, what are you getting at?
- Continue like that. Pick any four numbers amongst the numbers you already have to create a new number. The question is, can you ever reach 1000?
- I have a feeling that it is clearing up already.
Problem source: Wisconsin Mathematics, Science and Engineering Talent Search.
September 29th, 2009 at 2:17 am
By choosing the largest 4 terms each time, I get, successively, 10, 19, 36, 69, 134, 258, 497, 958. Then 958+36+4+2=1000.
September 29th, 2009 at 3:16 am
I interpret “pick any four” as meaning pick a subset consisting of four of the numbers you have, i.e. the four numbers added must be distinct. On this basis, you can reach every positive integer except:
* 5, 6, 7, 8, 9 which are too small to reach with four distinct terms
* 11, 12, 13, 14, 15 which are too large to reach with four distinct terms all less than 10, but too small if 10 is used
* 20, 21 which are too large to reach with four distinct terms none larger than 10, but too small if a term larger than 10 is used (that term must be at least 16=10+3+2+1).
You can reach every larger integer by successively adding 1+2+3 to any of:
10=4+3+2+1
17=10+4+2+1
18=10+4+3+1
19=10+4+3+2
26=19+4+2+1=18+3+2+1
27=19+4+3+1=18+4+3+2
For the largest number reachable in a given number of steps, see http://www.research.att.com/~njas/sequences/A145028
September 29th, 2009 at 3:24 am
Correction! 26=18+4+3+1.