# More Puzzles

 Mathematical Puzzles write 271 as the sum of positive real numbers
 Sat Dec 04, 2010 6:10 pm  by tartle write 271 as the sum of positive real numbers so as to maximize their product. Mon Jan 03, 2011 1:13 pm  by s.b. 2+3+4+5+6+7+8.... note:0 is not applied cuz then product becomes 0. 1 does not change the product...am i right? Sun May 29, 2011 7:48 pm  by bds021 135.5+135.5 Sat Jun 04, 2011 7:03 am  by Unni 3+2+2+2+2+2....... Product = 3 x 134th power of 2 Sat Jun 04, 2011 7:05 am  by Unni 3+2+2+2+2+2....... Product = 3 x 134th power of 2 Sun Jun 19, 2011 6:05 pm  by DiamondSoul It's 2.71 repeated 100 times. Wed Jul 06, 2011 4:28 pm  by cat [quote="DiamondSoul"]It's 2.71 repeated 100 times.[/quote] This appears to be correct, but needs a little explanation. ([i]n[/i]+1)*([i]n[/i]-1) = [i]n[/i]^2-1, which is less than [i]n[/i]^2. This shows that uniform values adding to a given sum make the largest product. Therefore, using [i]a[/i] for 271, the product [i]y[/i] of [i]x[/i] uniform values can be written as: [i]y[/i] = ([i]a[/i]/[i]x[/i])^[i]x[/i] = e^[[i]x[/i]*(ln[i]a[/i]-ln[i]x[/i])] The derivative is: [i]dy[/i]/[i]dx[/i] = [([i]a[/i]/[i]x[/i])^[i]x[/i]]*[ln[i]a[/i]-(1+ln[i]x[/i])] = [([i]a[/i]/[i]x[/i])^[i]x[/i]]*[ln([i]a[/i]/[i]x[/i])-1] At the maximum product, the derivative equals zero, so that: ln([i]a[/i]/[i]x[/i]) = 1; [i]a[/i]/[i]x[/i] = e; [i]x[/i] = [i]a[/i]/e = 271/2.718... = 100 to the nearest integer. All times are GMT Page 1 of 1