- --- In mathforfun@yahoogroups.com, clooneman@y... wrote:
>

2.7027027.... being very close to e.

> But if the addends could contain a decimal part, then they would

>all

> equal n/d, and you'd have:

> 2.7027027....*2.7027027....*2.7027027....*2.7027027.... (37 times)

> = 2.7027027....^37 = 9 474 061 716 781 832.651871822612603, as

>above.

>

2^11*3^26=5205741216417792

> P.S. Only one problem with my answer if n/d is an integer, because

> using 36 integers (three less 2s and two more 3's) yields 5 856 458

> 868 470 016. Whoops....

>

> P.P.S. And repeating that step again gets you a bigger product

> again....

2^8*3^28 =5856458868470016

2^5*3^30 =6588516227028768

2^2*3^32 =7412080755407364

Each new line is the previous divided by 8 and multiplied by 9.

>But interestingly, two 2's and thirty-two 3's gets you

duh. Because 2+2=4?

> exactly the same result as when you shrink the number of integers

>by

> 1 and use thirty-two 3's and one 4! Anyone see why?

This is interesting. When I did this problem before, I always used

rationals for addends. Rational addends close to e always gave the

best results. So when Peter used natural numbers for addends, I

thought they must be 3's and 2's because they are closest to e. Now

you are showing that using a 4 can give the same result. Four is not

very close to e.

btw

e^(100/e)=9479842689868732.4631217804012218

100/e=36.7879441171442321595523770161461

I can add e 36.7879.. times.

Can I multiply e 36.7879.. times?

>

(Phew......)^2

>

- --- In mathforfun@yahoogroups.com, "cooperpuzzles"

<cooperpuzzles@y...> wrote:>

More importantly 2x2=4

> --- In mathforfun@yahoogroups.com, clooneman@y... wrote:

> >

>

>

> duh. Because 2+2=4?

>

> This is interesting. When I did this problem before, I always used

not very close to e.

> rationals for addends. Rational addends close to e always gave the

> best results. So when Peter used natural numbers for addends, I

> thought they must be 3's and 2's because they are closest to e. Now

> you are showing that using a 4 can give the same result. Four is

The addends must be 2 or 3, not because they are closer to e, but

because an alternative to the addend 5, is the addends 2 and 3 which

have a product of 6, which is greater than 5

similarly 4 x 2 > 6, but 3x3 is even bigger.

4x3 >7

3x3x2>8

3x3x3>9

5x5 >10 and 3x3x4 is even bigger, because I showed above you would

not use the addend 5.

6x5>11, but you can do better than a 6 and a 5.

etc

so all addends greater than 4 would never be used.

4 only gets a run because 2x2=4

by using 4 you don't get a higher product, merely a smaller number

[count] of addends.

Peter