359 lines
14 KiB
Markdown
359 lines
14 KiB
Markdown
---
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title: Floors & Ceilings
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TARGET DECK: Obsidian::STEM
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FILE TAGS: algebra algorithm
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tags:
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- algebra
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---
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## Overview
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The **floor** of $x$ is the greatest integer less than $x$. The **ceiling** of $x$ is the least integer greater than $x$. These values are denoted $\lfloor x \rfloor$ and $\lceil x \rceil$ respectively.
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%%ANKI
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Basic
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How is the floor of $x$ denoted?
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Back: $\lfloor x \rfloor$
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708110779607-->
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END%%
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%%ANKI
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Basic
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What is the floor of $x$?
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Back: The greatest integer less than or equal to $x$.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708110779649-->
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END%%
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%%ANKI
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Basic
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How is the ceiling of $x$ denoted?
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Back: $\lceil x \rceil$
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708110779656-->
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END%%
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%%ANKI
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Basic
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What is the ceiling of $x$?
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Back: The least integer greater than or equal to $x$.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708110779663-->
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END%%
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%%ANKI
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Basic
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When does $\lfloor x / 2 \rfloor = \lceil x / 2 \rceil$?
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Back: When $x$ is even.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708110779668-->
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END%%
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%%ANKI
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Basic
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When does $\lfloor x / 2 \rfloor \neq \lceil x / 2 \rceil$?
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Back: When $x$ is odd.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708110779674-->
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END%%
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%%ANKI
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Basic
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What does $\lceil x \rceil - \lfloor x \rfloor$ equal?
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Back: Either $0$ or $1$.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708110779681-->
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END%%
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%%ANKI
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Basic
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What can be said about $x$ if $\lceil x \rceil - \lfloor x \rfloor = 0$?
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Back: $x$ is an integer.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708110779687-->
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END%%
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%%ANKI
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Basic
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What can be said about $x$ if $\lceil x \rceil - \lfloor x \rfloor = 1$?
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Back: $x$ is not an integer.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708110779693-->
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END%%
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%%ANKI
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Basic
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What values of $x$ satisfy $\lfloor x \rfloor = \lceil x \rceil - 1$?
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Back: Non-integral values.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1709831032489-->
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END%%
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%%ANKI
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Basic
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What values of $x$ satisfy $\lceil x \rceil = \lfloor x \rfloor + 1$?
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Back: Non-integral values.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1709831032496-->
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END%%
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%%ANKI
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Basic
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What values of $x$ satisfy $\lfloor x \rfloor = \lceil x \rceil + 1$?
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Back: None.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1709831032505-->
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END%%
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%%ANKI
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Basic
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What values of $x$ satisfy $\lceil x \rceil = \lfloor x \rfloor - 1$?
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Back: None.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1709831032513-->
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END%%
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%%ANKI
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Basic
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For what values $k$ is horizontal and vertical shifting of $\lfloor x \rfloor$ the same?
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Back: Integral values.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1709831032519-->
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END%%
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%%ANKI
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Basic
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What C operator corresponds to floor division?
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Back: N/A.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708110779699-->
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END%%
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%%ANKI
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Basic
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What C operator corresponds to integer division?
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Back: `/`
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Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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<!--ID: 1709831032524-->
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END%%
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%%ANKI
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Basic
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When does C operator `/` behave like floor division?
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Back: When the result is a nonnegative value.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708110779705-->
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END%%
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%%ANKI
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Basic
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When does C operator `/` behave like ceiling division?
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Back: When the result is a nonpositive value.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708110779710-->
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END%%
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%%ANKI
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Basic
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What C operator corresponds to ceiling division?
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Back: None.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708110779716-->
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END%%
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%%ANKI
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Basic
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How does C evaluate `10 / 3`?
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Back: `3`
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Tags: c17
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<!--ID: 1708110779720-->
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END%%
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%%ANKI
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Basic
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How does C evaluate `floor(10.f / 3)`?
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Back: `3`
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Tags: c17
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<!--ID: 1708110779725-->
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END%%
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%%ANKI
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Basic
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How does C evaluate `ceil(10.f / 3)`?
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Back: `4`
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Tags: c17
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<!--ID: 1708110779729-->
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END%%
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%%ANKI
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Basic
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How does C evaluate `-10 / 3`?
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Back: `-3`
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Tags: c17
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<!--ID: 1708110779734-->
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END%%
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%%ANKI
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Basic
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How does C evaluate `floor(-10.f / 3)`?
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Back: `-4`
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Tags: c17
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<!--ID: 1708110779738-->
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END%%
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%%ANKI
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Basic
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How does C evaluate `ceil(-10.f / 3)`?
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Back: `-3`
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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Tags: c17
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<!--ID: 1708110779742-->
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END%%
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%%ANKI
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Basic
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Given $r = \lfloor (p + q) / 2 \rfloor$, fair partitioning requires `A[r]` to be included in which of `A[p..r-1]` or `A[r+1..q]`?
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Back: `A[p..r-1]`
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708115109770-->
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END%%
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%%ANKI
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Basic
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Given $r = \lfloor (p + q) / 2 \rfloor$, when is `A[p..r]` or `A[r+1..q]` equally sized?
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Back: When `A[p..q]` has even size.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708115745322-->
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END%%
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%%ANKI
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Basic
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Given $r = \lceil (p + q) / 2 \rceil$, fair partitioning requires `A[r]` to be included in which of `A[p..r-1]` or `A[r+1..q]`?
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Back: `A[r+1..q]`
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708115109777-->
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END%%
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%%ANKI
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Basic
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If `A[p..q]` has odd size, what `r` most fairly allows partitions `A[p..r]` and `A[r+1..q]`?
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Back: $r = (p + q) / 2$
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708114757958-->
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END%%
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%%ANKI
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Basic
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If `A[p..q]` has odd size, what `r` most fairly allows partitions `A[p..r-1]` and `A[r..q]`?
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Back: $r = (p + q) / 2$
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708114757961-->
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END%%
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%%ANKI
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Basic
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If `A[p..q]` has odd size, what `r` ensures `A[p..r-1]` has same size as `A[r+1..q]`?
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Back: $r = (p + q) / 2$
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708114757964-->
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END%%
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%%ANKI
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Basic
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If `A[p..q]` has even size, what `r` most fairly allows partitions `A[p..r]` and `A[r+1..q]`?
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Back: $r = \lfloor (p + q) / 2 \rfloor$
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708114757968-->
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END%%
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%%ANKI
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Basic
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If `A[p..q]` has even size, what `r` most fairly allows partitions `A[p..r-1]` and `A[r..q]`?
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Back: $r = \lceil (p + q) / 2 \rceil$
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708114757971-->
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END%%
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%%ANKI
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Basic
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Given `A[p..q]` and $r = \lfloor (p + q) / 2 \rfloor$, how does the size of `A[p..r]` compare to `A[r+1..q]`?
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Back: It either has zero or one more members.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708115683351-->
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END%%
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%%ANKI
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Basic
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Given `A[p..q]` and $r = \lfloor (p + q) / 2 \rfloor$, what is the size of `A[p..r]` in terms of $n = q - p + 1$?
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Back: $\lceil n / 2 \rceil$.
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Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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<!--ID: 1708742467192-->
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END%%
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%%ANKI
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Basic
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Given `A[p..q]` and $r = \lfloor (p + q) / 2 \rfloor$, what is the size of `A[r+1..q]` in terms of $n = q - p + 1$?
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Back: $\lfloor n / 2 \rfloor$.
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Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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<!--ID: 1708742467198-->
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END%%
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%%ANKI
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Basic
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Given `A[p..q]` and $r = \lceil (p + q) / 2 \rceil$, how does the size of `A[p..r-1]` compare to `A[r..q]`?
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Back: It either has zero or one fewer members.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708115683354-->
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END%%
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%%ANKI
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Basic
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Given `A[p..q]` and $r = \lceil (p + q) / 2 \rceil$, what is the size of `A[r..q]` in terms of $n = q - p + 1$?
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Back: $\lceil n / 2 \rceil$.
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Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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<!--ID: 1708742467202-->
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END%%
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%%ANKI
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Basic
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Given `A[p..q]` and $r = \lceil (p + q) / 2 \rceil$, what is the size of `A[p..r-1]` in terms of $n = q - p + 1$?
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Back: $\lfloor n / 2 \rfloor$.
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Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
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<!--ID: 1708742467207-->
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END%%
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%%ANKI
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Basic
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Given `A[p..q]` and $r = \lfloor (p + q) / 2 \rfloor$, how does the size of `A[p..r-1]` compare to `A[r..q]`?
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Back: It either has one or two fewer members.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708115683358-->
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END%%
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%%ANKI
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Basic
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Given `A[p..q]` and $r = \lceil (p + q) / 2 \rceil$, how does the size of `A[p..r]` compare to `A[r+1..q]`?
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Back: It either has one or two more members.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708115683362-->
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END%%
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%%ANKI
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Basic
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Given `A[p..q]` and $r = \lfloor (p + q) / 2 \rfloor$, *why* is the size of `A[p..r]` potentially larger than `A[r+1..q]`?
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Back: If `A[p..q]` has odd size, `A[p..r]` contains the midpoint.
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Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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<!--ID: 1708115683366-->
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END%%
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## Bibliography
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* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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* Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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* Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). |