12.1: Recursion Basics
- Page ID
- 117599
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)By the end of this section you should be able to
- Describe the concept of recursion.
- Demonstrate how recursion uses simple solutions to build a better solution.
Recursion
Recursion is a problem solving technique that uses the solution to a simpler version of the problem to solve the bigger problem. In turn, the same technique can be applied to the simpler version.
How is the problem of moving two rings solved using recursion?
- Move the small ring to the middle tower, move the bigger ring to the target tower, and move the small ring to the target tower.
- Move two rings from the source tower to the target tower.
- Cannot be solved with recursion.
- Answer
-
a. The move of the smaller ring, which is a smaller problem, is done twice on the way to solving for the two rings overall. The task of moving all three rings is broken down into smaller tasks of moving one ring at a time.
How is the problem of moving three rings solved using recursion?
- Move three rings from the source tower to the target tower.
- Move two rings to the middle tower, then move the biggest ring to the target tower, and finally, move two rings to the target tower.
- Cannot be solved with recursion.
- Answer
-
b. The two-ring solution from question 1 is used in the solution for three rings.
How many times is the two-ring solution used with three rings?
- 1
- 2
- 0
- Answer
-
b. Once to move two rings to the middle tower following the rules, then to move two rings to the target tower.
Recursion to find a complete solution
The recursion process continues until the problem is small enough, at which point the solution is known or can easily be found. The larger solution can then be built systematically by successively building ever larger solutions until the complete problem is solved.
How many total steps does it take to solve two rings?
- 1
- 2
- 3
- Answer
-
c. The small ring is moved to the middle tower. The bigger ring is moved to the target tower. Then, the small ring is moved from the middle to the target.
How many total steps does it take to solve three rings?
- 3
- 4
- 7
- Answer
-
c. To solve, the two-ring solution is used twice, and there is one extra step to move the biggest ring from the source to the target. Each solution of two rings requires three steps each. So 3 + 1 + 3 = 7 steps total.
How many total steps does it take to solve four rings?
- 15
- 7
- 3
- Answer
-
a. The four-ring problem uses the three-ring solution twice. The three smaller rings are moved to the middle tower, taking seven steps as in question 2. One step is needed to move the biggest ring from the source to the target. It takes seven more steps move three smaller rings from the middle to the target. The total is 7 + 1 + 7 = 15 steps.