Solution to “The Greatest Number”

1) There are $9$ single digit numbers and $90$ two digit numbers $99-10+1$. $192$ digits in all. We have to remove $100$ of these digits, leaving behind a $92$ digit number.

2) The largest $92$ digit must have as many $9$s to begin with as possible. So, we drop the digits within the parenthesis – $(1...8)9(10...1)9(20...2)9(30...3)9(40...4)9$ to begin with. So, the number begins with $99999$. And we have eliminated $8 + 19x4 = 84$ digits.

3) The number we end with at the end of this step is $999995051525354...99100$. We will have to eliminate $16$ more digits.

4) Typically, we jump to the conclusion here. We could remove $50 51 52 53 54 55 56 57$ and conclude that the answer should be $99999585960...9899100$. This is wrong.

5) We need to remove the $5$s from $57 58$ instead. So, we eliminate the numbers within the parenthesis $(50 51 52 53 54 55 56 5)7(5)8$ to create the number $999997859606162....979899100$. This is the largest number possible.

5) Most of us get the general algorithm correct, but last minute twist makes it an absorbing exercise. Have fun !!!

The Greatest Number

The problems dealing with finding the largest number given a set of rules have been a great source of fun. We went through the analysis of the largest number that can be formed using four $2$s. We will now look at another problem or a puzzle,

• A number is formed by all digits from $1,2,3...100$ and in that sequence. Therefore our number is $123456789101112...9899100$.
• Now, we must delete 100 digits in all. Any 100 digits.
• What is the largest number that can be formed after you have removed the 100 digits, without shuffling the sequence of the digits that are left behind.
• Let us consider an example. The largest number that can be formed from $96785$ after removing two digits is $985$. We have removed two digits six and seven, but have retained the sequence of other digits that are left behind. This example would have helped you understand the rules of the game.
• Other questions to consider are: What are the digits you deleted and why?
• For starters, begin with asking a few basic questions. a) How many digits are there in the number to begin with? b) How many digits do we expect to have in the answer? c) What are the basic rules of arithmetic for creating the largest numbers?
• Most people rush towards the answer, forgetting to enjoy the journey of solving the problem. Learning is in the journey. The result is simply a consequence of how well you have applied what you have learned during the journey.

Have fun !!

What comes next ?

This is a good one. I found this one on the pages of The New Indian Express. The question was to determine the number that comes next in the sequence $100, 121, 144, 202, 244, \ldots$.

At first blush, I came to the conclusion, that the first three numbers are $10^2, 11^2, 12^2$ followed by $2 \times10^2+2 = 202, 2 \times 11^2+2 = 244$, therefore the next number must be $2 \times 12^2+2 = 290$.

This is a fascinating sequence if you use a bit of theory of notation of numbers. Can you arrive at the next number ?

Solution to the “A question of the biggest number”

What is the biggest number that can be formed using three 2s?

$2^{2^2}$ is smaller than $222$.  It is incorrect to assume that exponentiation always produces large numbers. And $22^2 = 484$. Therefore we have $2^{2^2} < 222 < 22^2 < 2^{22}$.  Therefore, the correct answer is $2^{22}$. We will have explain this a bit better, which we will in the lines that follow.

What is the biggest number that can be formed using three 3s?

We can extend the steps in the problem solving process like we did before.

$333 < 33^3 < 3^{3^3} < 3^{33}$.  Therefore  is the required answer is $3^{33}$. Let us see if this pattern can be generalized.

What is the biggest number that can be formed using three 4s?

This brings us to the question of determining the largest number with three 4s.

$444 < 44^4 < 4^{44} < 4^{4^4}$. Clearly, $4^{4^4} = 4^{256}$. This is greater than $4^{44}$.  The previous pattern is not working! We must explore what is causing this behavior. Why is $a^{a^a} < a^{aa}$, when $a = 1, 2, 3$ ?

The number in the exponent $aa$ in decimal number system, simply represents the following  $aa = 10a + a = 11a$.

Therefore the question under consideration can be represented as: When is $a^a < 11a$ or $a^{a-1} < 11$ ? This equation fails for all $a > 3$. This explains why the generation snapped when $a=4$.

Makes sense ? As always, post your comments, if you can think of a more logical way to explain your answers. The idea is to share our problem solving techniques.

A question of the biggest number – Part II

At the very outset, I would like to thank folks for posting their ideas and solutions.

Looks like I did not make thing clear in the previous puzzle. Let us assume that we are allowed to used the following operators – addition, subtraction, multiplication, division and exponentiation.

1. What is the biggest number with four 1s?
2. What is the biggest number with four 2s?

The idea behind these puzzles is to encourage children to think about exponentiation and multiplication operations. These questions also highlight how one cannot ‘generalize’ patterns, without also checking if these generalizations work,  while arriving at the solution. Most of us use intelligent guesswork as a strategy. I would like to request you to refrain from this, and actually reason out why you believe that your answer is correct. The idea here is to inculcate the habit of working systematically and deriving facts, one from the other.

Have fun !

A question of the biggest number

This is a fascinating puzzle. It is easy to solve and it is equally easy to make an error in judgment and arrive at the wrong answer.

1. Which is the largest number that can be formed using three 2s ?
2. Which is the largest number that can be formed using three 3s ?
3. Which is the largest number that can be formed using three 4s ?

Hint: There is a good reason why we are asking you to find the answer for three consecutive numbers.

Post your responses, we will review the solution in some detail in the next post.

Handling Math

How do we create a Math  blog  which can handle equations seamlessly ? This is a question that haunts folks who want to create any article or tutorial on Math. The classical approach has been to create small jpg with the equations in them and embedding them in HTML to create a Math artifact. Or use MathType in MSWord and export the document into HTML; and this would seamlessly accomplish the same thing.

After looking for various solutions, I believe that we have a clean solution to this problem. Thanks to wordpress.com, we can now use LaTeX to typeset our articles and online documents. I came across this blog that outlined the way to make this happen. A couple of tries – and I was up and running.

In fact, this blog has been the prime motivation for me to work on a math-blog. There are alternatives like the one in this blog . But you should be aware of its limitations. It also requires MathPlayer. For additional details on this implementation, you may review this link.

I switched to WordPress from blogger to exploit the ease with which I can embed expressions like $\displaystyle \lim_{x_0 \to -\infty} e^{sin x}$ in no time. So I can focus on my message and content, rather than worrying about the gazillion workarounds to make this happen.

I wondered how WordPress could handle this ? The answer lies in MathJax. The way to use this capability is to do the following:

1. Insert the following line in your HTML code.<script type=”text/javascript” src=”http://cdn.mathjax.org/mathjax/1.1-latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML-full”></script&gt;
2. Now you can type your equations like you would do in LaTeX, with a minor difference.  You would embed the equation between $latex and$, of a pair of $. 3. For example dollar latex x^2 + 2x + 1 =0 dollar would produce $x^2 + 2x + 1 = 0$. I have written “dollar” instead of$ to make things readable.
4. You do not have to download MathJax on to your machine. But you will have to stay online to get this to work.