Eyes Down
The symbol [ ] means 'the integer part of'. Can the numbers [2x]; 2[x]; [x + 1/2] + [x - 1/2] ever be equal? Can they ever take three different values?
Problem
The symbol [ ] means 'the integer part of '.
Consider the three numbers
$$[2x];\ 2[x];\ [x + {1\over 2}] + [x - {1\over 2}]$$
Can they ever be equal?
Can they ever take three different values?
Getting Started
If the integer part of $x$ is $a$ then $x=a + b$ where $a$ is a whole number and $0\leq b < 1$.
Try some numerical values of $x$, evaluate the three functions and record the results. What do you notice? Can you prove that different values of $x$ will produce similar findings?
Student Solutions
Thank you to Alan of Madras College for this solution.
If $x$ is a real number then $x = a + b$ where $a$ is an integer and $b$ is a real number such that $0 \leq b < 1$. Here $a$ is the integer part of $x$ and we write $a = [x]$. We have to consider whether $[2x]$; $2[x]$ and $[x + 1/2 ] + [x - 1/2 ]$ can ever be equal and whether they can take three different values.
If $1/2 \leq b < 1$ then $[2x]= 2a + 1$.
If $0 \leq b < 1/2$ then $[2x]= 2a$.
For any $b$, $2[x] = 2a$.
If $1/2 \leq b < 1$ then $[x+ 1/2 ] = a + 1$ and $[x - 1/2 ] = a$ and so $[x + 1/2 ] + [x - 1/2 ] = 2a + 1$.
If $0 \leq b < 1/2$ then $[x+ 1/2 ] = a$ and $[x - 1/2 ] = a - 1$ and so $[x + 1/2 ] + [x - 1/2 ] = 2a - 1$.
$\bullet$ Case 1: $\; 0 \leq b < 1/2$
$[2x]= 2a = 2[x]$
but $[2x] \neq [x + 1/2 ] + [x - 1/2 ]$.
$\bullet$ Case 2: $\; 1/2 \leq b < 1$
$[2x]= 2a + 1 = [x + 1/2 ] + [x - 1/2]$
but $[2x] \neq 2[x]$.
Hence it is impossible for all of $[2x]$; $2[x]$ and $[x + 1/2 ] + [x - 1/2 ]$ to be equal but they can never take three different values.
Teachers' Resources
Why do this problem?
The problem gives practice in working with linear inequalities and in working systematically through separate cases.
Possible approach
Encourage the class to try some numerical values for $x$, to compare the values of the three functions and to record their findings. Collect sufficient results from the class to provide evidence for spotting patterns and making conjectures.
Key questions
If the integer part of $x$ is $a$ then $x=a + b$ where $a$ is a whole number and $0\leq b < 1$. What is the difference between the separate cases where $0 \leq b < {1\over 2}$ and ${1\over 2}\leq b < 1$?