This is a brain teaser.

maychang

This is a brain teaser. [Copy link]

 

okhxyyo seems to like to post brain teasers. Now I will post one on behalf of okhxyyo: How to represent three decimal numbers without using operators as the largest integer? Three 1s: 1 times 1 is still 1, and 1 to the power of 1 is still 1. Obviously, the largest integer that can be formed by three 1s without using operators is Three 9s: The largest integer that can be formed by three 9s is not . In fact, the largest integer that can be formed by two 9s without using any arithmetic operators is 360837, which is 387420489 in decimal, much larger than 999. Three 9s can be expressed as 360839, which is a 95-digit decimal number. Three 9s can be expressed as 360840, which is an incredibly large integer. Here comes the question: @okhxyyo(1) How many digits does 360849 have when written as a decimal number? (2) How to express three numbers 2, three numbers 3, three numbers 4, etc. as the largest integer without using any operation symbols? Adapted from "Fun Algebra" by Barry Lehrman

star_66666

1. Is it an 81-digit number? 2. 222 is the largest, 3 ((3)3), 4 is the same, that is, after 3, it is exponential, and the data is the largest.

maychang

star_66666 posted on 2018-6-26 11:37 1. Is it an 81-digit number? 2. 222 is the largest, and so is 3 ((3)3). The same is true for 4, which means that after 3, the data is the largest in exponential form.

22 to the power of 2 is 484, and 2 to the power of 22 is 4194304, both of which are larger than 222. 3 to the power of 3 is 27, and 3 to the power of 33 is obviously larger than 3 to the power of 27.

okhxyyo

9 to the power of 9 is a 9-digit number, which is very interesting. The first digit is 987654321.9 to the power of 9, which is 81 digits.

star_66666

There are still many tall people.

maychang

okhxyyo posted on 2018-6-26 12:16 9 to the power of 9 is 9 digits, very interesting, the first digit is 987654321.9 to the power of 9, 81 digits.

9 to the power of 9, 81 digits. No way? , written in decimal, is 387420489. , is a 95-digit decimal number. , written in decimal, can only have more digits than 95, not 81. is 9 raised to the power of 387420489.

okhxyyo

maychang posted on 2018-6-26 12:47 9 to the 9th power of 9 is 81 digits. No way? , written in decimal form is 387420489. , is a 95-digit decimal number...

⊙⊙! Let me think about it calmly

maychang

okhxyyo posted on 2018-6-26 12:51 ⊙⊙! Let me think about it calmly

Let me think about it calmly Hint: Take the logarithm of 9 to the power of 387420489 (any base).

maychang

okhxyyo posted on 2018-6-26 12:53 I didn't calculate, and didn't count the carry at the end, 囧囧囧, but what does 95 decimal mean, isn't this a 9-digit number?

But what does 95 decimal mean, isn't this a 9-digit number? is equal to 2.951266543 multiplied by 10 to the 94th power, which is a 95-digit decimal number.

AOE-hill

Is this a cramp question? I got cramps right after I finished the calculation!

wsmysyn

1. Take the common logarithm, which should be (9^9 * lg9 + 1) decimal digits (369, 693, 100 digits?) 2. 2^22, 3^33 are the largest, and the rest are 4^(4^4) which is the largest. .

maychang

wsmysyn posted on 2018-6-26 14:33 1. Taking the common logarithm, it should be (9^9 * lg9 + 1) so many decimal digits (369,693,100 so many digits?) 2. 2^22, 3^33 is the most...

Indeed. More than 300 million digits.

wsmysyn

maychang posted on 2018-6-26 15:06 Indeed. More than 300 million digits.

When I saw this number, although it was written in a simple way, the number of digits was already more than 300 million, and then I thought of the Ge Liheng number

maychang

wsmysyn posted on 2018-6-26 15:21 When I saw this number, although it is written in a simple way, the number of digits is more than 300 million, and then I thought of the Graham number

Compared with the Graham number, this more than 300 million digits is just zero, just 9^9^9.

wsmysyn

maychang posted on 2018-6-26 15:43 Compared with the Graham number, this 300 million digits is just a zero, 9^9^9.

Yes, yes... It seems that there are known meaningful numbers that are much larger than the Graham number... It's already a headache to understand it.

chunyang

It is actually not difficult to answer such questions as long as you have a spirit of questioning and empiricism, and questioning and empiricism are the foundation of science.