logo programming book preview

Contents

Section I: Basics of Logo ....................................................... 11 1. Getting Started with Logo .......................................................................... 13

2. Drawing with the Logo Turtle ................................................................... 23

3. Instructions and Procedures....................................................................... 38

4. Teaching New Words.................................................................................. 45

5. Designs with Repeating Patterns............................................................... 61

Section II: Powerful Ideas..................................................... 73 6. Control through Input................................................................................. 75

7. Drawing Polygons and Circles .................................................................. 87

8. Painting Our Designs ................................................................................ 105

9. Mirror Images, Flowers, and other Designs........................................... 115

10. Problem‐solving using Divide and Conquer ......................................... 133

Section III: Appendices....................................................... 153 Appendix A: Logo Primitives (Built‐in Procedures) .................................... 155

Appendix B: Installing Logo ............................................................................ 160

Contents

Section IV: Advanced Exploration........................................ 7

11. Drawing Curvy Objects ................................................................................ 9

12. Power of Polygons ....................................................................................... 35

13. Perspectives and Concentric Shapes ......................................................... 57

Section V: Programming Adventures................................. 77

14. Fooled by RANDOM................................................................................... 79

15. Adding Life ‐ Introduction to Animation................................................. 91

16. Turtles that Climb Trees............................................................................ 105

17. Looking Back and Final Project................................................................ 129

Section VI: Appendices....................................................... 133

Appendix C: Additional Logo Primitives (Built‐in Procedures) ................ 135

Series on Learning through Computer Programming 5

Preface The other day, while working on this book, one of the authors was struggling to get a feature in his word processor to work. His 8‐year old son was watching curiously from behind to see what his Dad was up to on his shining new laptop. When he noticed Dadʹs hesitation, he leaped over his Dadʹs shoulders, clicked a few places on the screen and exclaimed, ʺThere you go, Dad! Is that what you wanted?ʺ Dad was naturally very pleased with his son, and also amazed at his sonʹs knowledge of this not‐so‐common feature.

The truth is: Children have a natural fascination for computers. We see young children ‐ some of them barely out of preschool ‐ handling with ease computers and gadgets such as video‐game consoles and mobile devices. They browse the Internet, connect with friends using social networks, play games, or use ready‐made software like word processors. Our schools have surely begun to realize the promises of the digital revolution. Yet, the best attempts in many schools to introduce the computer to their students do not go beyond treating it as a productivity tool, i.e. as an assistant for routine work. What really needs to happen is that our children take the driverʹs seat, feel the real power of computers, do creative work, and learn in the process. This book demonstrates a novel idea that can achieve this exact goal.

The new idea is to use Programming as a medium for learning. Programming means tapping into the computerʹs immense power by talking with it directly. Through programming, children use the computerʹs terrific power to draw graphics, design animation, solve mathematical or word puzzles, and even build robots. This idea was first proposed in the famous book ʺMindstormsʺ by Seymour Papert and has subsequently been appreciated and praised by educators and parents all over the world.

Fun with Logo Programming – Part 1

6 Series on Learning through Computer Programming

Children see the computer as a powerful assistant. Through programming, students discover that the computer is a powerful and flexible tool. Using interesting ideas embedded in programming environments, students solve problems in their favorite subjects, and also develop interest in ʺdifficultʺ subjects like Math and Physics.

Children learn through programming. • Children become active learners, because they explore ideas through a hands‐

on activity. • They learn to think about and analyze their own thinking, because that is the

only way to program computers. • They learn the technique of dealing with complex problems by breaking

them into smaller sub‐problems. • They learn a new way of thinking (called ʺcomputationalʺ thinking). • In the world of programming, answers are not simply ʺrightʺ or ʺwrongʺ; this

prepares childrenʹs mindset for real life problems. • Childrenʹs overall learning process transforms from acquiring facts to creative

and analytical thinking.

References 1. Mindstorms: Children, Computers, and Powerful Ideas by Seymour Papert,

Professor at MIT. (Also http://papert.org/: Papertʹs website) 2. Computer Science Logo Style by Brian Harvey, Professor at UC Berkeley. 3. Apple Logo for Teachers by Earl Babbie (Wadsworth Publishers, 1984) 4. http://www.cs.berkeley.edu/~bh/: Articles by Prof. Brian Harvey, UC

Berkeley. 5. http://el.media.mit.edu/logo-foundation/: Website of Logo Foundation 6. http://scratch.mit.edu: Website of the Scratch project


1. Getting Started with Logo

Why Learn Logo Programming? Computer programming is one of the most exciting and powerful ways to use a computer. And Logo is a fascinating programming language that practically anyone ‐ children and adults alike ‐ can learn to explore the art of computer programming. You may ask: what is so special about programming? Let us look at the picture below.

Figure 1‐1

The following Logo program draws this figure: REPEAT 6 [FORWARD 100 RIGHT 60]



You can also easily draw it using drawing software like Microsoft Paint. So, whatʹs the difference? Why should anyone bother to learn programming when readymade software can do the job? Well, now, see the picture below. By modifying your Logo program just a little you can create this new picture. REPEAT 6 [ REPEAT 6 [FORWARD 50 RIGHT 60] RIGHT 60 ]

Figure 1‐2

Now, try drawing that using Microsoft Paint!! This is just one example of how Programming gives you a special power that no ready‐made software can. Programming gives you a direct hotline to the brain of the computer. It allows you to try out your creative ideas. And most of all, it is a lot of fun!


2. Drawing with the Logo Turtle

Introduction In the previous chapter, we got a glimpse of the power of Logo programming. We started talking with the computer using words like PRINT. Can we get Logo to draw things on the screen, like the ones shown below?

Figure 2‐1

The answer is Yes! We can create these and many other exciting drawings with the help of the Logo Turtle! Let us get introduced to this new friend.

Drawing with the Logo Turtle


? RIGHT 90

Figure 2‐4

Using the command RIGHT 90, we told the Turtle to turn right by 90 degrees. Now, we can ask the Turtle to move forward again by typing the following. ? FORWARD 100

Figure 2‐5 This is getting interesting, isnʹt it! Can you complete the picture and draw a square? Here is how we can draw the remaining sides of the square and bring the Turtle back to its original position. ? RIGHT 90 ? FORWARD 100 ? RIGHT 90 ? FORWARD 100 ? RIGHT 90

Figure 2‐6

What if we wanted to erase everything on the screen and start all over? Simple! We have done it once already. Type: ? CLEARSCREEN

Teaching New Words


4. Teaching New Words

Introduction We are now able to create interesting shapes and designs using words (procedures) that Logo knows. These procedures are called Logo primitives. Wouldn’t it be fun if we could teach Logo new words? For example, it would be nice if Logo could draw the blocks (as shown below), when we simply type blocks: ? blocks

Figure 4‐1

In this chapter, we will see how we can teach Logo such new words. Right now, of course, Logo will just give us the error message: I don’t know how to blocks.

Teaching New Words


The Concept Called “Abstraction” (Just FYI) Abstraction means capturing an idea into a word (or phrase). The new procedure “Square” is an abstraction of the idea of drawing a shape consisting of 4 sides. Defining a new word that has a specific meaning is called abstraction. No big deal!

Play Time: Define the following new Logo procedures. Verify that these new procedures work as expected. 1. A procedure called “Myinfo” that clears the screen and shows your full name,

school name, and city. 2. Procedures for geometric shapes like rectangle and triangle. 3. Procedures for English letters (stick or block) like i, H, E, or A. 4. Procedure for rectangle wave.

Logo Challenge Can you write a Logo procedure to draw this design?

Figure 4‐6

New words and old words: Once defined, a new procedure is treated (by Logo) just like any other procedure that Logo already knows. There is no special treatment for new procedures. For example, if you want to define yet another new procedure, you can make use of both old (primitives) and new procedures.


5. Designs with Repeating Patterns

Introduction Take a look at the following design. It appears like a powerful multi‐color star.

Figure 5‐1

Do we know enough of Logo to draw this star? Probably not. There is actually a powerful new way to draw this star (and many other exciting designs) quite easily. Read on to learn about it.

Designs with Repeating Patterns


How about drawing a rectangle wave?

Figure 5‐4

Do you see a repeating pattern in each of these? Just like the staircase, there is a basic pattern (that is repeated) in each of the above pictures. If we can work out the Logo instructions for each basic pattern, we can simply type these instructions a number of times to get the final picture. So, for each of these pictures a certain set of instructions needs to be repeated. Fortunately, instead of us having to type these instructions repeatedly, we can simply use a procedure that Logo provides to perform the repetition for us. Can you guess the name of that Logo procedure? Yes! You guessed correctly. It is called REPEAT.

How to Use the REPEAT Procedure Ask Logo to say “I am an idiot” 4 times! ? REPEAT 4 [PRINT [I am an idiot]] I am an idiot I am an idiot I am an idiot I am an idiot The basic format of REPEAT procedure is as follows: REPEAT N [List of Instructions] N is the number of times you want some actions to be repeated, and List of Instructions gives the actions that you want to be repeated.



Drawing a star using REPEAT: Ok, you think that REPEAT saves typing effort. And you are right. But, REPEAT is much more interesting than just being a ʺtyping assistantʺ. Drawing a star (as below) gives us a perfect opportunity to glimpse the power of REPEAT. Letʹs say we want to draw the ʺpoint starʺ as shown here. It is basically a bunch of rays (line segments) coming out of a point. Right? This particular star has 20 rays. And the gap (angle) between each adjacent pair of rays is constant.

Figure 5‐9

Letʹs see how we would go about drawing such a star in Logo. To simplify the problem, let us first try a star with fewer rays. This star has 8 rays. We will assume that the Turtle is initially at the center. After drawing each ray (FD and BK), the Turtle has to take a right turn. Since there are 8 equal turns, each of these turns (angles) must be 45 (360/8).

Figure 5‐10



What We Learnt in this Chapter 1. When we draw designs that have a repeating pattern, it gets tedious to type the

same instructions again and again. REPEAT comes to rescue. 2. Introduction to REPEAT:

a. REPEAT is a labor‐saving idea ‐ not for the computer, but for us. b. It is also a powerful idea ‐ something that opens up new possibilities.

3. We begin to see the value of using a text editor (like Notepad) ‐ it keeps a record of programs AND helps in debugging (trouble‐shooting).

4. The key sequence CNTL-Q helps you get the ʺ?ʺ prompt (ALT-S on UCB Logo 6.0) if you want to terminate a program, or if you don’t see the ʺ?ʺ prompt for any reason.

5. It is important to first figure out (in your mind or notebook) what exactly is getting repeated before using REPEAT.

Suggestions to Teachers/Parents REPEAT is an extremely powerful idea. It will help to practice it to really appreciate its power and to understand how best to utilize it. Think of repetitive patterns and designs which can be drawn using the basic Turtle commands, and give them to the children for practice.


6. Control through Input

Introduction With the ideas we have learnt so far ‐ Turtle commands, creating new procedures, and the REPEAT command ‐ can we draw some of the designs shown below?

Figure 6‐1

We probably can. We can use our ‘Point.Star’ procedure to draw the stars. But, we will have to constantly modify the procedure to draw the variety shown above. Logo offers an idea that allows a single star procedure to draw any and all of the stars above. Won’t that be interesting? So, read on this chapter to learn this new idea.

Control through Input


Step 4 is to put this name at the end of the TO line as shown below. This tells Logo that size is the input of Hexagon. TO Hexagon :size Step 5 (and the final step) is to put the new name wherever we refer to size of the hexagon. After making all these changes this is what we get: ERASE "Hexagon TO Hexagon :size REPEAT 6 [FD :size RT 60] END Note: Donʹt forget the colon ‐ it tells Logo that the word that follows is an input. Without ʺ:ʺ Logo will treat the word as procedure name. Bingo! Our hexagon procedure will now take input just like FD or RT. ? Hexagon not enough inputs to hexagon With inputs we get:

Hexagon 100

Figure 6‐3

Hexagon 50

Figure 6‐4


8. Painting Our Designs

Introduction Drawing figures (outlines) usually goes hand in hand with painting, i.e. filling areas with color. And of course, color makes our creative ideas more lively and exciting.

Figure 8‐1

In this chapter, we will learn how to paint shapes in Logo, i.e. how to fill color in our designs to make them look beautiful. And it is quite straightforward. You can then draw pretty designs as shown above.



And then the jump and paint job:

PU RT 45 FD 10 PD

SETPC 2 FILL

Since the nearest closed shape for the Turtle (when FILL is called) is the space between the two squares, thatʹs what got painted.

Figure 8‐8

Play Time: 1. Write a procedure called fsquare that takes 2 inputs: size and color, and

draws a painted square. For example, ʺfsquare 100 2" will draw a square of size 100 painted green. ʺfsquare 300 13" will draw a square of size 300 painted purple (Solution explained in the following section).

2. Figure out how you can paint a triangle. Write a procedure ftriangle with size and color inputs.

The FSQUARE Procedure We know that the following set of instructions draws a square of size 100 and paints it with the red color. ; Draw a square REPEAT 4 [FD 100 RT 90] ; Jump inside PU RT 45 FD 50 PD ; Set color to red and paint SETPC 4 FILL To finish the job, we should get the Turtle back to its original position. ; Jump back outside PU BK 50 LT 45 PD We can put all these instructions into a procedure called FSQUARE as below:


9. Mirror Images, Flowers, and other Designs

Introduction Using a basic shape (such as a simple square), we can create a lot of entertaining designs by moving it in different ways. Imagine that you have cut out a shape drawn on paper. You could draw its outline by placing it anywhere on a drawing paper. Or, you could flip it along an edge to get a mirror image. Or, you can hold one of its corners with a real pin and rotate it around this pin, while drawing outlines all along to get a flowery pattern. We will learn how to do these cool things using simple techniques in Logo.

Hopping Around This is the simplest of the possible movements. See below:

Figure 9‐1



Nested REPEAT (Just FYI): Instead of using the square procedure, if we had simply used Logo instructions, our program to draw a flower of 18 squares would look as follows: REPEAT 18 [ REPEAT 4 [FD 100 RT 90] RT 20 ] We have used a REPEAT inside another REPEAT. In programming lingo, this is called nested REPEAT. Just for your information!

Since we are calling our designs as flowers, we will call each of its squares as a petal.

Play Time: 1. Write a procedure called sqflower (short for flower of squares) that takes 2

inputs: size and a number N: size is for the size of each square, and N specifies the number of petals. So, for example: sqflower 100 5 should draw a flower of 5 squares each of size 100 sqflower 50 20 should draw a flower of 20 squares each of size 50

2. Replace the square with a triangle and draw various types of triangle flowers: dumb‐bell (2 petals), fan (4 petals), and so on.

Figure 9‐14

Figure 9‐15

Figure 9‐16

3. Draw flowers of other polygon shapes: pentagon, hexagon, etc.



Key Observations • REPEAT is not just a handy little command that saves typing; amazing new

possibilities open up that would be impossible without REPEAT. (Imagine drawing a flower of circles without the help of REPEAT).

• Before using REPEAT, you must first figure out the repeated basic pattern, which may consist of a visible pattern (lines) and an invisible pattern (angles and gaps).

• In order to figure out the basic pattern in a complicated design, simply trace the motion of Turtle from the central point of the design and note all lines and turns. Remember, in a regular polygon, all lines and turns must be equal.

• Programming allows infinite experimentation ‐ so donʹt be afraid to try out your ideas and hunches. If they turn out to be incorrect (weird stuff gets drawn on the screen), either pat yourself on the back for amazing new discoveries (if the weird stuff looks pretty) or go back to your notebook, make changes, and try again.

• Do not forget to solve the problem in your mind (and notebook) first, before writing Logo programs. Remember that Logo is not going to help you solve the problem; it will simply replicate all the mistakes that you have made yourself in your design.

• The geometry you learnt in school does come handy in drawing designs. In turn, Logo programming will help you understand and appreciate your school geometry.

• When you define a new Logo procedure, define it without any input first. Make sure it works as expected. And then add inputs one at a time, testing the procedure at every step.


10. Problem-solving Using Divide and Conquer

Introduction We have been learning a lot of little tricks of programming so far and applying them to write lots of little programs. Now, we are ready to write some ʺseriously coolʺ programs. See the collage below.

Figure 10‐1

But, before you start churning out Logo programs for these fancy objects, letʹs take a couple of simple examples and understand the process of how to go about developing such programs.



that there is very little room for making mistakes, and the components can be used for other designs also.

Self‐study: Using the above procedure, how will you draw block arrows pointing in other directions, such as, North or South?

Play Time: Draw a collage of block arrows of different sizes, pointing in different directions, and of different colors and thickness.

House Example Letʹs go through another interesting example to make sure we understand the Divide‐and‐conquer method. We will attempt to draw the house as shown here. First, we take a step back and take a keen look at the design and ask the standard question:

Figure 10‐8 ʺWhat is this design made up of? Can we break it apart into meaningful reusable components or parts?ʺ We come up with the answer: ʺYes, we see three parts: a wall, a roof, and a window.ʺ

Figure 10‐9


11. Drawing Curvy Objects

Introduction With the exception of circles, we have so far only seen how to draw designs that contain straight lines. We don’t know yet how to draw nice curvy objects like a lotus flower, a flying bird, or a fish. Or a beautiful garden as shown below:

Figure 11‐1

In this chapter, we will discover that we can draw curves without needing to learn anything new; not even new Logo commands. We will see that itʹs all just the magic of simple geometry.

Drawing Curvy Objects


The median of the petal would be at half the angle of the right turn (i.e. 45). So, in order to get a symmetric petal around the Turtleʹs present orientation, we should tilt the Turtle to the left by 45 before drawing the petal. ERASE "petal2 TO petal2 :size ; size is diameter (of the full circle) LT 45 REPEAT 2 [ q.dcircle :size RT 90 ] RT 45 END Petal2 150

Figure 11‐11

Play Time: 1. Draw a candle as shown below.

Figure 11‐12

Drawing Curvy Objects


;Quarter circle anti-clockwise. Size is diameter. ERASE "Q.dcircleL TO Q.dcircleL :size REPEAT 90 [FD :size*3.14159/360 LT 1] END Using a combination of right quarter circle and left quarter circle we can create a nice looking wave: ERASE "waves TO waves :n :size REPEAT :n [ q.dcircle :size q.dcircleL :size ] END waves 3 40

Figure 11‐26

Self‐study: Make the wave above appear horizontal.

Logo Challenges

Chestnut leaves:

Figure 11‐27

The figure shows the leaf pattern of a Chestnut tree. The amazing thing is that all Chestnut leaves appear in the exact same pattern as shown in Figure 11‐27: always in groups of 7 with one large leaf at the top, and remaining 6 appearing as mirror images. Hint: First write a procedure to draw an individual leaf (use a ʹsizeʹ input to control the size of the leaf). Then, design the pattern above.


12. Power of Polygons

Introduction Earlier, we used our knowledge of geometry to draw interesting but simple designs using regular polygons. We can explore this idea further and develop some expertise in this very exciting field of “Geometric Designs”. You must have seen patterns ‐ like the one shown here ‐ used for tiles, cutlery, architectures, and even paintings. All that we really need to know to create such designs is polygons, Turtle Round Trip principle, and the Logo commands that we already know. Are you interested? Then, read on.

Figure 12‐1



Reversing Sides What will we need to do to draw a figure shown below?

Figure 12‐9

What if you were told that this figure is related to the next figure in some way? Clearly, the outside octagon has been made to hide. But, how do we get the following figure in the first place?

Here is the code that will produce the design. REPEAT 8 [ ;Draw the inside triangle Polygon 100 3 ;Draw side of octagon FD 100 RT 360/8 ] It’s really the same idea as before, except this time, the triangle is drawn on the inside of the main octagon.

Figure 12‐10 You could get rid of the octagon in two ways: (1) avoid drawing it in the first place – this will require you to modify the triangle procedure, or (2) Use PENERASE to erase the Octagon. The following code shows the second approach to get the star shown above. PENERASE REPEAT 8 [FD 100 RT 360/8] PENPAINT


13. Perspectives and Concentric Shapes

Introduction Do you like the design shown below?

Figure 13‐1

Can you guess how it can be drawn in Logo? Yes! You can give a series of circle commands and increase the size slightly every time. But, that would be tedious, wonʹt it?



How about drawing this beautiful design shown next?

Figure 13‐2

Can you guess how this design can be drawn in Logo? If you look carefully, you will see a series of squares drawn around a point, with each square slightly bigger than the previous one. Once again, you could give a series of square commands and increase the size slightly every time. But, that would be horribly tedious, wouldnʹt it! Continue reading this chapter, if you would like to learn a much more elegant and simpler way of drawing this sort of designs in which there is repetition for sure, but there is a twist after each repetition.

Counting Repetitions Imagine that you want to run around a tree, and your friend is watching you standing aside. You could ask him to count 1, 2, 3… as you go around the tree. So, you could ask him any time, “Hey, what round am I doing now?” And he would tell you, ʺItʹs the 9th roundʺ, or something like that. In a sense, your friend is your counter.


14. Fooled by RANDOM

Introduction One of the most exciting ideas in programming, and even in real life, is the idea of randomness. We often say things like, ʺOh, that person is so random!ʺ Philosophers often muse about the randomness of events. The industry of gambling is entirely based on the idea of random outcomes. Take a look at the picture below:

Figure 14‐1

Fooled by RANDOM


Figure 14‐4

As you can see, we get this weird‐looking window. In fact, your window will most likely look quite different! Get it? It’s the RANDOM thing!

Now, if we simply put this in a loop and run it a number of times, we get something that looks like a shattered window! REPEAT 50 [ REPEAT 4 [ square (RANDOM 150) RT 90 ] ]

Figure 14‐5

If you increase the outer loop count to a really large number (like 10000) you will just get a large painted square. That’s because, when run 10000 times, RANDOM ends up giving practically all possible numbers in the available range (0 to 100), and so, we get squares of all possible sizes drawn very close to each other giving the effect of FILL.

Insight: If called a large number of times, RANDOM usually ends up giving every value in the given range – although this is not guaranteed, nor is it predictable how often it will give a particular value.

So, as you see, the application of RANDOM really depends on our creativity.


15. Adding Life - Introduction to Animation

Introduction Animation means life. Animation means motion and action. Animation means fun! The good news is that we can do animation in Logo. So far, our Logo programs have produced nice‐looking graphics and images. We are now ready to make our Logo drawings come alive and perform actions. As you might know already, animation is an illusion of motion (or action) created by persistence of vision and a successive display of slowly changing pictures. For a quick demo of animation, just run the following program and watch the screen. Press Ctrl‐Q (Alt‐S in UCB Logo 6.0) when you have seen enough! REPEAT FOREVER [ FD 100 BK 100 ; Draw a line WAIT 5 ; Wait for some time PENERASE ; Turn the Eraser mode on FD 100 BK 100 ; Erase the line drawn earlier PENPAINT ; Restore the Pen RT 5 ; Turn right by 5 degrees ] Isnʹt it interesting? Can you figure out how the program works by reading the comments in the code? (Note: REPEAT FOREVER is a special form of REPEAT in which things go on, well, forever!) Read further to learn how to write animation programs in Logo.



ERASE "Bcwheel TO Bcwheel SETPENSIZE 5 REPEAT 100 [ FD 50 BK 50 RT 360/100 ] SETPENSIZE 20 REPEAT 10 [ PU FD 50 FD 20 PD FD 50 BK 50 PU BK 20 BK 50 PD RT 360/10 ] END CS SETBG 7 SETPC 0 Bcwheel

Figure 15‐7

Shapes with Wings and Legs Well, we donʹt mean literally, but in the sense that our stationary shapes will start moving around the screen as if they were living or motorized objects. In a previous chapter, we wrote a procedure to draw a block arrow as shown below. Now, we will see how we can make it move around the screen.

Figure 15‐8

To create the moving arrow effect, we could try the following steps: 1. Draw a block arrow. 2. Wait for a short time. 3. Clear the screen.


16. Turtles that Climb Trees

Introduction You probably noticed the Fern branch and Tropical tree drawn on the very first page of this book. Here they are once more:

Figure 16‐1

Believe it or not, these figures have been drawn using Logo programs – programs containing not more than 15 instructions each! They are based on an interesting idea called Recursion. Continue reading this chapter to learn about recursion and how to draw these sort of designs in Logo.

Turtles that Climb Trees


Hint: The following instructions will draw a single curved line and return the Turtle to its base (this code would actually be level 1 of the recursion): REPEAT 10 [FD 25 RT 5] REPEAT 10 [LT 5 BK 25] You just have to recursively implant the same curved line at a number of places on both sides.

Logo Challenge The trees shown next have the following new features: - The tree looks different every time you call the instruction (even if all inputs are

the same) - The tree shows fruit at some places. The fruit shows up at random places, and

the amount of fruit is controlled through an input. Hints: (1) Obviously, we have used RANDOM to make things happen differently every time. (2) To draw the fruit, add code just before the STOP command, because that’s where the branches end.

Figure 16‐18

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