# Most consecutive positive integers using two 1s

Using two 1s, try to come up with the most consecutive positive integers.

Allowed operations:

• Subtraction

• Multiplication

• Division

• Concatenation

• Square Root

• Factorial

• Floor and Ceiling Functions

• Decimal Point

• Isn't there a conjecture that with only factorial, square root and floor, and starting with any single number >1, you can form all positive integers? If so, this seems like one that could go on for ever... – Gareth McCaughan Jan 23 '17 at 3:52
• @GarethMcCaughan It'd have to be $>2$, because $2!=2$, floor decreases and square root makes closer to 1. – boboquack Jan 23 '17 at 4:34
• @GarethMcCaughan ... and floor yields an integer. – Rosie F Jan 23 '17 at 7:04
• Oops, yes, I meant >2. – Gareth McCaughan Jan 23 '17 at 10:04
• Do you happen to recall the name of this conjecture? – greenturtle3141 Jan 24 '17 at 0:48

Let's try (feel free to add on or correct, this is community wiki):

$1=1\times1$

$2=1+1$

$3=\left\lfloor\sqrt{11}\right\rfloor$

$4=\left\lceil\sqrt{11}\right\rceil$

$5=\left\lfloor\sqrt{\sqrt{\left(\left\lfloor\sqrt{11}\right\rfloor!\right)!}}\right\rfloor$

$6=\left\lfloor\sqrt{11}\right\rfloor!$

$7=\left\lceil\sqrt{\sqrt{11}!}!\right\rceil$

$8=\left\lfloor\sqrt{\sqrt{\sqrt{11!}}}\right\rfloor$

$9=\left\lfloor\sqrt{11}!\right\rfloor$

$10=\left\lceil\sqrt{11}!\right\rceil$

$11=11$
$12=\left\lceil\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\left\lceil\sqrt{\sqrt{\sqrt{\sqrt{(\left\lceil\sqrt{11}\space\right\rceil!)!}}}}\right\rceil!}}}}}\right\rceil$
$13=\left\lceil\sqrt{\sqrt{\sqrt{\left\lceil\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\left\lceil\sqrt{\sqrt{\sqrt{\sqrt{(\left\lceil\sqrt{11}\space\right\rceil!)!}}}}\right\rceil!}}}}}\right\rceil}}}\right\rceil$

• I found a way for 12 but it's just a hassle to write it down. We can go all the way to infinity without much problem. – stack reader Jan 23 '17 at 6:05
• Getting past 10 was the hard part. – humn Jan 23 '17 at 6:08
• @humn I simplified my 10 as well as the 7 and 9. Looks much better now. But I guess 11 is a good number to stop this... I am not going any higher lol. – stack reader Jan 23 '17 at 6:17
• Anyone want to write a program for this...? – greenturtle3141 Jan 23 '17 at 6:19
• This is turning into a ridiculous open-ended mathematics exercise. It seems likely that given an unlimited number of factorials, square roots, floor and ceilings, you could express the values of all positive integers. – wildBillMunson Jan 23 '17 at 13:46