5 - Functions.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Functions\n",
    "\n",
    "Topics:\n",
    "1. How to declare a function\n",
    "2. Duck-typing in Julia\n",
    "3. Mutating vs. non-mutating functions\n",
    "4. Some higher order functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## How to declare a function\n",
    "Julia gives us a few different ways to write a function. The first requires the `function` and `end` keywords"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sayhi (generic function with 1 method)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function sayhi(name)\n",
    "    println(\"Hi $name, it's great to see you!\")\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "f (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function f(x)\n",
    "    x^2\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can call either of these functions like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Hi C-3PO, it's great to see you!\n"
     ]
    }
   ],
   "source": [
    "sayhi(\"C-3PO\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1764"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(42)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Alternatively, we could have declared either of these functions in a single line"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sayhi2 (generic function with 1 method)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sayhi2(name) = println(\"Hi $name, it's great to see you!\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "f2 (generic function with 1 method)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f2(x) = x^2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Hi R2D2, it's great to see you!\n"
     ]
    }
   ],
   "source": [
    "sayhi2(\"R2D2\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1764"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f2(42)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we could have declared these as \"anonymous\" functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "#1 (generic function with 1 method)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sayhi3 = name -> println(\"Hi $name, it's great to see you!\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "#3 (generic function with 1 method)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f3 = x -> x^2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Hi Chewbacca, it's great to see you!\n"
     ]
    }
   ],
   "source": [
    "sayhi3(\"Chewbacca\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1764"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f3(42)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Duck-typing in Julia\n",
    "*\"If it quacks like a duck, it's a duck.\"* <br><br>\n",
    "Julia functions will just work on whatever inputs make sense. <br><br>\n",
    "For example, `sayhi` works on the name of this minor tv character, written as an integer..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Hi 55595472, it's great to see you!\n"
     ]
    }
   ],
   "source": [
    "sayhi(55595472)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And `f` will work on a matrix. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Float64}:\n",
       " 0.983045  0.336857  0.21284\n",
       " 0.102728  0.711431  0.817376\n",
       " 0.504421  0.681324  0.833934"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = rand(3, 3)\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Float64}:\n",
       " 1.10834   0.715809  0.662064\n",
       " 0.586371  1.09764   1.28501\n",
       " 0.986512  1.22281   1.3597"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`f` will also work on a string like \"hi\" because `*` is defined for string inputs as string concatenation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"hihi\""
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(\"hi\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On the other hand, `f` will not work on a vector. Unlike `A^2`, which is well-defined, the meaning of `v^2` for a vector, `v`, is not a well-defined algebraic operation. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Float64}:\n",
       " 0.1488599996403983\n",
       " 0.5795662436976401\n",
       " 0.12786532477145474"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = rand(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "ename": "LoadError",
     "evalue": "MethodError: no method matching ^(::Vector{Int64}, ::Int64)\n\u001b[0mClosest candidates are:\n\u001b[0m  ^(\u001b[91m::Union{AbstractChar, AbstractString}\u001b[39m, ::Integer) at strings/basic.jl:718\n\u001b[0m  ^(\u001b[91m::LinearAlgebra.Symmetric{var\"#s832\", S} where {var\"#s832\"<:Real, S<:(AbstractMatrix{var\"#s832\"} where var\"#s832\"<:var\"#s832\")}\u001b[39m, ::Integer) at /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.6/LinearAlgebra/src/symmetric.jl:868\n\u001b[0m  ^(\u001b[91m::LinearAlgebra.Symmetric{var\"#s832\", S} where {var\"#s832\"<:Complex, S<:(AbstractMatrix{var\"#s832\"} where var\"#s832\"<:var\"#s832\")}\u001b[39m, ::Integer) at /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.6/LinearAlgebra/src/symmetric.jl:869\n\u001b[0m  ...",
     "output_type": "error",
     "traceback": [
      "MethodError: no method matching ^(::Vector{Int64}, ::Int64)\n\u001b[0mClosest candidates are:\n\u001b[0m  ^(\u001b[91m::Union{AbstractChar, AbstractString}\u001b[39m, ::Integer) at strings/basic.jl:718\n\u001b[0m  ^(\u001b[91m::LinearAlgebra.Symmetric{var\"#s832\", S} where {var\"#s832\"<:Real, S<:(AbstractMatrix{var\"#s832\"} where var\"#s832\"<:var\"#s832\")}\u001b[39m, ::Integer) at /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.6/LinearAlgebra/src/symmetric.jl:868\n\u001b[0m  ^(\u001b[91m::LinearAlgebra.Symmetric{var\"#s832\", S} where {var\"#s832\"<:Complex, S<:(AbstractMatrix{var\"#s832\"} where var\"#s832\"<:var\"#s832\")}\u001b[39m, ::Integer) at /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.6/LinearAlgebra/src/symmetric.jl:869\n\u001b[0m  ...",
      "",
      "Stacktrace:",
      " [1] macro expansion",
      "   @ ./none:0 [inlined]",
      " [2] literal_pow",
      "   @ ./none:0 [inlined]",
      " [3] f(x::Vector{Int64})",
      "   @ Main ./In[2]:2",
      " [4] top-level scope",
      "   @ In[22]:2",
      " [5] eval",
      "   @ ./boot.jl:360 [inlined]",
      " [6] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)",
      "   @ Base ./loading.jl:1094"
     ]
    }
   ],
   "source": [
    "# This won't work\n",
    "f(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Mutating vs. non-mutating functions\n",
    "\n",
    "By convention, functions followed by `!` alter their contents and functions lacking `!` do not.\n",
    "\n",
    "For example, let's look at the difference between `sort` and `sort!`.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Int64}:\n",
       " 3\n",
       " 5\n",
       " 2"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = [3, 5, 2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Int64}:\n",
       " 2\n",
       " 3\n",
       " 5"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sort(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Int64}:\n",
       " 3\n",
       " 5\n",
       " 2"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`sort(v)` returns a sorted array that contains the same elements as `v`, but `v` is left unchanged. <br><br>\n",
    "\n",
    "On the other hand, when we run `sort!(v)`, the contents of v are sorted within the array `v`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Int64}:\n",
       " 2\n",
       " 3\n",
       " 5"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sort!(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Int64}:\n",
       " 2\n",
       " 3\n",
       " 5"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Some higher order functions\n",
    "\n",
    "### map\n",
    "\n",
    "`map` is a \"higher-order\" function in Julia that *takes a function* as one of its input arguments. \n",
    "`map` then applies that function to every element of the data structure you pass it. For example, executing\n",
    "\n",
    "```julia\n",
    "map(f, [1, 2, 3])\n",
    "```\n",
    "will give you an output array where the function `f` has been applied to all elements of `[1, 2, 3]`\n",
    "```julia\n",
    "[f(1), f(2), f(3)]\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Int64}:\n",
       " 1\n",
       " 4\n",
       " 9"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "map(f, [1, 2, 3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we've squared all the elements of the vector `[1, 2, 3]`, rather than squaring the vector `[1, 2, 3]`.\n",
    "\n",
    "To do this, we could have passed to `map` an anonymous function rather than a named function, such as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "#5 (generic function with 1 method)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x -> x^3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "via"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Int64}:\n",
       "  1\n",
       "  8\n",
       " 27"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "map(x -> x^3, [1, 2, 3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and now we've cubed all the elements of `[1, 2, 3]`!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### broadcast\n",
    "\n",
    "`broadcast` is another higher-order function like `map`. `broadcast` is a generalization of `map`, so it can do every thing `map` can do and more. The syntax for calling `broadcast` is the same as for calling `map`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Int64}:\n",
       " 1\n",
       " 4\n",
       " 9"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "broadcast(f, [1, 2, 3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and again, we've applied `f` (squared) to all the elements of `[1, 2, 3]` - this time by \"broadcasting\" `f`!\n",
    "\n",
    "Some syntactic sugar for calling `broadcast` is to place a `.` between the name of the function you want to `broadcast` and its input arguments. For example,\n",
    "\n",
    "```julia\n",
    "broadcast(f, [1, 2, 3])\n",
    "```\n",
    "is the same as\n",
    "```julia\n",
    "f.([1, 2, 3])\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Int64}:\n",
       " 1\n",
       " 4\n",
       " 9"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.([1, 2, 3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice again how different this is from calling \n",
    "```julia\n",
    "f([1, 2, 3])\n",
    "```\n",
    "We can square every element of a vector, but we can't square a vector!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To drive home the point, let's look at the difference between\n",
    "\n",
    "```julia\n",
    "f(A)\n",
    "```\n",
    "and\n",
    "```julia\n",
    "f.(A)\n",
    "```\n",
    "for a matrix `A`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Int64}:\n",
       " 1  2  3\n",
       " 4  5  6\n",
       " 7  8  9"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = [i + 3*j for j in 0:2, i in 1:3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Int64}:\n",
       "  30   36   42\n",
       "  66   81   96\n",
       " 102  126  150"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As before we see that for a matrix, `A`,\n",
    "```\n",
    "f(A) = A^2 = A * A\n",
    "``` \n",
    "\n",
    "On the other hand,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Int64}:\n",
       "  1   4   9\n",
       " 16  25  36\n",
       " 49  64  81"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = f.(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "contains the squares of all the entries of `A`.\n",
    "\n",
    "This dot syntax for broadcasting allows us to write relatively complex compound elementwise expressions in a way that looks natural/closer to mathematical notation. For example, we can write"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Float64}:\n",
       "  3.0   6.0   9.0\n",
       " 12.0  15.0  18.0\n",
       " 21.0  24.0  27.0"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A .+ 2 .* f.(A) ./ A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "instead of"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Float64}:\n",
       "  3.0   6.0   9.0\n",
       " 12.0  15.0  18.0\n",
       " 21.0  24.0  27.0"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "broadcast(x -> x + 2 * f(x) / x, A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and the two will perform exactly the same."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercises\n",
    "\n",
    "#### 6.1 \n",
    "Write a function `add_one` that adds 1 to its input."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
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    "editable": false,
    "hide_input": true,
    "nbgrader": {
     "checksum": "253b17dc2f3d3a58042fbc36042a0fd5",
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   },
   "outputs": [],
   "source": [
    "@assert add_one(1) == 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "editable": false,
    "hide_input": true,
    "nbgrader": {
     "checksum": "4e05440e19cd3606df11186d41d562bf",
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     "locked": true,
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     "schema_version": 1,
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    }
   },
   "outputs": [],
   "source": [
    "@assert add_one(11) == 12"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 6.2 \n",
    "Use `map` or `broadcast` to increment every element of matrix `A` by `1` and assign it to a variable `A1`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "@assert A1 == [2 3 4; 5 6 7; 8 9 10]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 6.3 \n",
    "Use the broadcast dot syntax to increment every element of matrix `A1` by `1` and store it in variable `A2`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "editable": false,
    "hide_input": true,
    "nbgrader": {
     "checksum": "3e3d797962df904deed0e7ee7782b69a",
     "grade": true,
     "grade_id": "cell-f3bd5479679a8fe1",
     "locked": true,
     "points": 0,
     "schema_version": 1,
     "solution": false
    }
   },
   "outputs": [],
   "source": [
    "@assert A2 == [3 4 5; 6 7 8; 9 10 11]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Please click on `Validate` on the top, once you are done with the exercises."
   ]
  }
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