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Difference between revisions of "2007 iTest Problems/Problem 41"

(Solution to Problem 41 — exhausting and satisfying NT problem)
 
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==Solution==
 
==Solution==
  
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To begin, we can find out for what values of <math>n</math> result in a certain <math>f(n)</math> (or <math>m</math>).  A table can help organize the findings.
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To begin, we can find out for what digits of the string is part of an <math>m</math>-digit number.  A table can help organize the findings.
  
 
{|class="wikitable" style="margin-left: auto; margin-right: auto;"
 
{|class="wikitable" style="margin-left: auto; margin-right: auto;"
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|<math>m</math>
 
|<math>m</math>
 
|Number of digits that make up <math>m</math>-digit numbers
 
|Number of digits that make up <math>m</math>-digit numbers
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|Highest value of <math>n</math> where <math>f(n) = m</math>
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|Highest value of <math>p</math> where the <math>p^\text{th}</math> digit of the string is part of an <math>m</math>-digit number
 
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|<math>1</math>
 
|<math>1</math>
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|}
 
|}
  
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It seems as if for a given value <math>m</math>, highest <math>n</math> where <math>f(n) = m</math> is a string of digits with <math>n-1</math> followed by <math>n-1</math> eights and a nine.  To prove that this is the case, we use [[induction]].  The base case for both of our wanted outcomes is seen in the table.
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It seems as if for a given value <math>m</math>, highest <math>p</math> where the <math>p^\text{th}</math> digit of the string is part of an <math>m</math>-digit number is a string of digits with <math>n-1</math> followed by <math>n-1</math> eights and a nine.  To prove that this is the case, we use [[induction]].  The base case for both of our wanted outcomes is seen in the table.
  
 
<br>
 
<br>
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For the inductive step, assume the highest <math>n</math> where <math>f(n) = m</math> is the string of digits <math>(m-1) \underbrace{888 \cdots 8}_{m-1 \text{ digits}} 9</math>
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For the inductive step, assume the highest <math>p</math> where the <math>p^\text{th}</math> digit of the string is part of an <math>m</math>-digit number is the string of digits <math>(m-1) \underbrace{888 \cdots 8}_{m-1 \text{ digits}} 9</math>.  The number of digits that make up <math>(m+1)</math> digit numbers is <math>((10^{m+1} - 1) - (10^{m} + 1)) \cdot (m+1) = 10^{m} \cdot 9(m+1)</math>, which is the string of digits <math>9(m+1)</math> followed by <math>m</math> zeroes.
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There are <math>((10^{m+1} - 1) - (10^{m} + 1)) \cdot (m+1) = 10^{m} \cdot 9(m+1)</math> (the string of digits <math>9(m+1)</math> followed by <math>m</math> zeroes) digits that make up <math>(m+1)</math> digit numbers.
 
  
 
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<br>
 
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Using the new discovery, the highest <math>n</math> where <math>f(n) = 2007</math> has <math>4 + 2006 + 1 = 2011</math> digits, the highest <math>n</math> where <math>f(n) = 2003</math> has <math>4 + 2002 + 1 = 2007</math> digits, and the highest <math>n</math> where <math>f(n) = 2004</math> has <math>4 + 2003 + 1 = 2008</math> digits.  Since <math>10^{2007}</math> has <math>2008</math> digits and <math>10^{2007}</math> is less than <math>2003\underbrace{888 \cdots 8}_{2003 \text{ digits}} 9</math> (highest <math>n</math> where <math>f(n) = 2004</math>), <math>f(10^{2007}) = \boxed{2004}</math>.
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Using the new discovery, the highest <math>p</math> where the <math>p^\text{th}</math> digit of the string is part of an <math>2007</math>-digit number has <math>4 + 2006 + 1 = 2011</math> digits, the highest <math>p</math> where the <math>p^\text{th}</math> digit of the string is part of an <math>2003</math>-digit number has <math>4 + 2002 + 1 = 2007</math> digits, and the highest <math>p</math> where the <math>p^\text{th}</math> digit of the string is part of an <math>2004</math>-digit number has <math>4 + 2003 + 1 = 2008</math> digits.  Since <math>10^{2007}</math> has <math>2008</math> digits and <math>10^{2007}</math> is less than <math>2003\underbrace{888 \cdots 8}_{2003 \text{ digits}} 9</math> (highest <math>p</math> where the <math>p^\text{th}</math> digit of the string is part of an <math>2004</math>-digit number), <math>f(2007) = \boxed{2004}</math>.
  
 
==See Also==
 
==See Also==

Latest revision as of 18:22, 2 July 2018

Problem

The sequence of digits $123456789101112131415161718192021\ldots$ is obtained by writing the positive integers in order. If the $10^{nth}$ digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2) = 2$ because the $100^{\text{th}}$ digit enters the sequence in the placement of the two-digit integer $55$. Find the value of $f(2007)$.

Solution

To begin, we can find out for what digits of the string is part of an $m$-digit number. A table can help organize the findings.

$m$ Number of digits that make up $m$-digit numbers Highest value of $p$ where the $p^\text{th}$ digit of the string is part of an $m$-digit number
$1$ $9$ $9$
$2$ $180$ $189$
$3$ $2700$ $2889$
$4$ $36000$ $38889$
$5$ $45 \cdot 10^4$ $488889$

It seems as if for a given value $m$, highest $p$ where the $p^\text{th}$ digit of the string is part of an $m$-digit number is a string of digits with $n-1$ followed by $n-1$ eights and a nine. To prove that this is the case, we use induction. The base case for both of our wanted outcomes is seen in the table.


For the inductive step, assume the highest $p$ where the $p^\text{th}$ digit of the string is part of an $m$-digit number is the string of digits $(m-1) \underbrace{888 \cdots 8}_{m-1 \text{ digits}} 9$. The number of digits that make up $(m+1)$ digit numbers is $((10^{m+1} - 1) - (10^{m} + 1)) \cdot (m+1) = 10^{m} \cdot 9(m+1)$, which is the string of digits $9(m+1)$ followed by $m$ zeroes.


Note that the last $m$ digits of $10^{m} \cdot 9m$ are zeroes, so the last $m$ digits are still $\underbrace{888 \cdots 8}_{m-1 \text{ digits}} 9$. The preceding string of digits are $m-1+9m+9 = 10m + 8$. The first string of digits are the string of digits of $m$, and it is followed by $m$ eights and a nine, so the inductive step is complete.


Using the new discovery, the highest $p$ where the $p^\text{th}$ digit of the string is part of an $2007$-digit number has $4 + 2006 + 1 = 2011$ digits, the highest $p$ where the $p^\text{th}$ digit of the string is part of an $2003$-digit number has $4 + 2002 + 1 = 2007$ digits, and the highest $p$ where the $p^\text{th}$ digit of the string is part of an $2004$-digit number has $4 + 2003 + 1 = 2008$ digits. Since $10^{2007}$ has $2008$ digits and $10^{2007}$ is less than $2003\underbrace{888 \cdots 8}_{2003 \text{ digits}} 9$ (highest $p$ where the $p^\text{th}$ digit of the string is part of an $2004$-digit number), $f(2007) = \boxed{2004}$.

See Also

2007 iTest (Problems, Answer Key)
Preceded by:
Problem 40 		Followed by:
Problem 42
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • TB1 • TB2 • TB3 • TB4

Solution

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