Two squirrels, A and B are talking about another squirrel, C.
A: Getting ready for winter is hard! I just gathered up another acorn.
B: I am a slow worker, so I am doing what I did last year and collecting one acorn per day. I got mine for today already.
A: Did you hear how many acorns C has now?
B: Yes, he has the product of how many I had yesterday, today, and will have tomorrow after I have gathered that day's acorn.
A: Anyone who has that many acorns must be a hard worker. I heard he has the product of my and your current acorn quantities, minus their sum.
If both squirrels were correct in their statements, how many acorns does each squirrel have (assume each squirrel has a whole number of acorns).
Let the number each has today be given by a, b, c respectively.
B's statement about C tells us c=(b-1)*b*(b+1)=b^3-b
A's statement tells us c=a*b-a-b
Setting these equal we have b^3=a(b-1). Subtract 1 from each side and
divide by b-1. (If b=1 then c=0 AND c=-1 which is impossible).
This gives us b^2+b+1=a-1/(b-1). Since the left side is an integer, the
right side must also be, so b=2. Solving for a and c we get:
(After writing this up, I saw that Ady had posted essentially the same thing. But since I'd gone to all that effort...perhaps the detail will enlighten someone?)
Edited on February 12, 2004, 4:29 pm