Question 10300
Let's work this one backwards:

Mary ends up with 3 donuts after giving half of them and a half a donut to her sister Kathy.  This means that before giving donuts to her sister Kathy, Mary must have had 7 donuts (of the 7 donuts, she gave half of them--3 1/2--plus a half, which would be 4 donuts to her sister, leaving her with 3 donuts for herself!)


Now, this means that she ended up with 7 donuts after giving half of them plus a half donut to her aunt.  By the same thinking, she must have had 15 donuts before giving 8 donuts to her aunt (7 1/2 + 1/2), leaving 7 for herself.


So, if she ended up with 15 donuts after giving half plus a half a donut to her mother, she must have started out with 31 donuts.


Now check it by seeing if it works:


Mary started with 31 donuts (half would be 15 1/2, + 1/2 means that she gave 16 to her mother), which leaves her with 15.  She gave 7 1/2 + 1/2 or 8 to her aunt, leaving her with 7 donuts.  She gave 3 1/2 + 1/2 or 4 to her sister, leavning her with  3 donuts or 1/4 of a dozen.  


Do you think there is a math formula for this?  Did you notice that at each step of the problem, each time she had donuts left, it was 1 less than a power of 2?  She started with 31 ( 32 -1), went down to 15 (16-1), then to 7 (8-1), and 3 (4-1).  Interesting problem.  Probably some high powered math here!!


R^2 at SCC