13x666=2020
 

In honor of UncWilly's New Math:
[url]https://www.mersenneforum.org/showpost.php?p=564288&postcount=1068[/url]

Find a formula using only 13 and 666 to get 2020.
You can only use +. -, x, /, ^, !, √

1. Find an answer with the fewest numbers (not digits).
2. Find an answer with the same number of each; and the fewest.

For example my first attempt uses 11 numbers.
So its an answer for 1. but not for 2.
666+666+666+13+13-(13+13)/13-(666+666)/666.

13*13*13-13*13-(13+13+13+13+13+13+13+13)/13 = 2020

[QUOTE=Viliam Furik;564344]13*13*13-13*13-(13+13+13+13+13+13+13+13)/13 = 2020[/QUOTE]

13*(13*13-13)-(13+13+13+13+13+13+13+13)/13 = 2020

13 13's

[QUOTE=Viliam Furik;564345]13*(13*13-13)-(13+13+13+13+13+13+13+13)/13 = 2020

13 13's[/QUOTE]

So next I'll expect 666 666s?

In the spirit of the original formula, solutions should use both 13 and 666.

Trivial improvement to the original solution - 10 numbers

666+666+666-(666+666+666+666)/666+13+13

Not much of a record but the first to use ^
(13+13+13+13)/13 + (13+13/13)*(13-13/13)^((13+13)/13)


7 of each
(666+666+666+666)/666 + (13+666/666)*(13-13/13)^((13+13)/13)

Trial and error got me down to 9 numbers:



2020 = 666+((13+13)/13)*(666+13-(13+13)/13)

[QUOTE=henryzz;564355]Not much of a record but the first to use ^
(13+13+13+13)/13 + (13+13/13)*(13-13/13)^((13+13)/13)


7 of each
(666+666+666+666)/666 + (13+666/666)*(13-13/13)^((13+13)/13)[/QUOTE]
The second one is really a smooth answer!

((666+666)/666)**(13-(13+13)/13)-13-13-(666+666)/666


6 each.

[C]**[/C] means [C]^[/C]——I was using python as my calculator, as many would have guessed it.

Written with [C]^[/C], my answer would be [QUOTE]((666+666)/666)^(13-(13+13)/13)-13-13-(666+666)/666[/QUOTE]

[QUOTE=Ensigm;564375]The second one is really a smooth answer![/QUOTE]

I realized the first answer could have half its 13s converted to any number. In fact, only 2 need to be 13s.

Using 2^11 was a nice solution.

9 of each using factorials
((666+666+666)/666)!^((13+13+13+13)/13)+666+13+13+13+13+((666+666+666)/666)!

Hitting blank trying to use sqrt

[QUOTE=henryzz;564386]I realized the first answer could have half its 13s converted to any number. In fact, only 2 need to be 13s. [/QUOTE]Aha, I didn't realize the two answers are essentially the same until now.

In saying "smooth" I was making a pun about the fact that the answer makes good use of the 7-smoothness of 2016.

[QUOTE=henryzz;564386]Hitting blank trying to use sqrt[/QUOTE]

is SQRT(666*666) cheap?

2020 = 666[SUB]13[/SUB] + 666[SUB]13[/SUB] - 13*13 - (13+13+13+13+13+13+13)/13
2 ea 666
12 ea 13
total 14

2020 = 13#/((666+666+666)/666) - (666 x 13) + 666 + (13+13)/13
6 ea 666
5 ea 13
total 11

[QUOTE=Uncwilly;564402]2020 = 666[SUB]13[/SUB] + 666[SUB]13[/SUB] - 13*13 - (13+13+13+13+13+13+13)/13
2 ea 666
12 ea 13
total 14[/QUOTE]


3C3+3C3= ???

666÷6.66×(13+13)÷1.3+(666+666)÷6.66

Getting a little inventive.. but unc started it ..Nya Nya

[QUOTE=Uncwilly;564402]2020 = 666[SUB]13[/SUB] + 666[SUB]13[/SUB] - 13*13 - (13+13+13+13+13+13+13)/13
2 ea 666
12 ea 13
total 14[/QUOTE]
666[SUB]13[/SUB] + 666[SUB]13[/SUB] - 13*13 - (13+13+13+13+13+13+13)/13
1098[SUB]10[/SUB] + 1098[SUB]10[/SUB] - 169 - 91/13
2196 - 169 - 7
2027 - 7
2020

13*13*13 - 13*13 - ((666+666)/666)^((666+666+666)/666)

666 + 666 + 666 + ((13+13)/13)*(13 - (13+13)/13)

I had checked the idea of the stated equation being valid in some base. This led to a cubic equation whose only real root was about -5.09.

I concluded that the "little know fact" was an "alternative fact."

Does 2020.000008 with 6 numbers count? :razz: (almost certainly beatable btw)

[code]sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(666!)))))))))*sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(666!)))))))))))))*sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(666!)))))))))))))))))))))))))))))*sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(13)))))))))))))))))))))/sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(666!))))))))))))))))))))))/sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(13))))))))))))))[/code]

Theoretically it should be possible to get arbitrarily close to 2020 using only one number by nesting factorials as well as sqrts...

Certainly not the smallest, but has some symmetry and uses 5 operators, +,-,^,/,()

((((((13+13)/13)^((13+13)/13)+(13/13))+(((666+666)/666)^((666+666)/666)))*(((13+13)/13)^((13+13)/13)+(13/13)))^((666+666)/666))-(((13+13)/13)^((13+13)/13)+(13/13))


i.e. 2^2+1=5, 2^2=4, 5+4=9, 9*5=45, 45^2 = 2025, 2025-5=2020

[QUOTE=petrw1;564401]is SQRT(666*666) cheap?[/QUOTE]

Was trying to avoid that

(666 + (666+666+666)/666) * (13+13+13)/13 + 13

Uses 5 of each.

13 * (13*13+13-666/(13+13)) - 13

8 numbers used.

[QUOTE=petrw1;564343]
Find a formula using only 13 and 666 to get 2020.
You can only use +. -, x, /, ^, !, √

1. Find an answer with the fewest numbers (not digits).
2. Find an answer with the same number of each; and the fewest.

[/QUOTE]

For Q2 the optimal is 9 terms (using 4 pieces of 13 and 5 pieces 666):
[CODE]
13-(666-(666+666)*(666+13*(13+13)))/666
[/CODE]
and for Q1 the optimal is the above 8 terms.
Assuming that we would use "only" the +,-,*,/ operations (no other crazy . or concat) and no large (intermediate) numbers.

I've proven by brute force search that there does not exist a solution using either 7 numbers or 4 of each, if we only allow the simple operators +, -, *, /.

The closest solution with 4 of each is 13 + 13 + 666 + 666 + 666 - 666/(13*13) = 2020.05917...

2020 = 666 + 666 + 666 + sqrt(666 - 13*13 - 13)

7 numbers.

[QUOTE=swishzzz;564498]2020 = 666 + 666 + 666 + sqrt(666 - 13*13 - 13)

7 numbers.[/QUOTE]

Genius!

[QUOTE=swishzzz;564498]2020 = 666 + 666 + 666 + sqrt(666 - 13*13 - 13)

7 numbers.[/QUOTE]

Nice!

[QUOTE=swishzzz;564498]2020 = 666 + 666 + 666 + sqrt(666 - 13*13 - 13)

7 numbers.[/QUOTE]

Very impressive

Note this can be converted to 4 of each quite easily:
2020 = 666 + 666 + 666 + sqrt(666 - 13*(13+13/13))

It would surprise me if these are beaten

[QUOTE=swishzzz;564498]2020 = 666 + 666 + 666 + sqrt(666 - 13*13 - 13)

7 numbers.[/QUOTE]
[QUOTE=henryzz;564568]
Note this can be converted to 4 of each quite easily:
2020 = 666 + 666 + 666 + sqrt(666 - 13*(13+13/13))

It would surprise me if these are beaten[/QUOTE]
I can only marvel at these.

[QUOTE=henryzz;564568]Very impressive

Note this can be converted to 4 of each quite easily:
2020 = 666 + 666 + 666 + sqrt(666 - 13*(13+13/13))

It would surprise me if these are beaten[/QUOTE]

Got to love it.

The newspapers need to be told - the World needs to know :alien:

I suggest the Daily Mail or the Sun in the UK, and Fox News for the telly. And the boys and girls in QAnon, and the anti Vaxxers, and the Covid deniers, and, and....

[QUOTE=robert44444uk;564661][QUOTE=henryzz;564568]Very impressive

Note this can be converted to 4 of each quite easily:
2020 = 666 + 666 + 666 + sqrt(666 - 13*(13+13/13))

It would surprise me if these are beaten[/QUOTE]
Got to love it.

The newspapers need to be told - the World needs to know :alien:

I suggest the Daily Mail or the Sun in the UK, and Fox News for the telly. And the boys and girls in QAnon, and the anti Vaxxers, and the Covid deniers, and, and....[/QUOTE]Late-breaking news! In the "art of beasting" department, the number of letters in the President's first, middle, and last names are 6, 4, and 5. Now, calculate

6*4*5 + 546

It's simple arithmetic! Numbers don't lie! What more needs to be said?

[size=1]Maybe this thread has run its course[/size]

[QUOTE=robert44444uk;564661]Got to love it.

The newspapers need to be told - the World needs to know :alien:

I suggest the Daily Mail or the Sun in the UK, and Fox News for the telly. And the boys and girls in QAnon, and the anti Vaxxers, and the Covid deniers, and, and....[/QUOTE]
Nature needs to hold the presses.