Note: In this problem, you’re asked to enter some APL code, including an APL symbol and the data (text or numbers) given to it. Your code will be tested by the APL Challenge system to check whether it gives the correct answer and uses the methods described below.

1: Drop

The symbol ↓ is used to drop items (the symbol hints at the direction that things fall when they are dropped). It takes a number on its left and a list on its right. The number specifies how many items to drop from the start of the list. For example, 1 ↓ 3 1 4 1 5 gives 1 4 1 5 and 3 ↓ 3 1 4 1 5 gives 1 5.

In APL, a text is a list of letters and must be between single quotes ( ' ). For example, 'ABC' will give a result that looks like ABC.

Write some APL code that uses ↓ to turn the text 'allocation' into 'location'.

In this problem, you’re asked to enter some APL code, including an APL symbol and the data (text or numbers) given to it. Your code will be tested by the APL Challenge system to check whether it gives the correct answer and uses the methods described below.

2: Meet Me at the Intersection

The symbol ∩ is called Intersection (its shape is like the letter n of intersection). It takes one list on each side and gives a new list. The new list will have every item from the list on the left that is also found in the list on the right. For example, 3 1 4 1 5 ∩ 1 2 3 gives 3 1 1 because:

• 3 is in the new list because it is found in 1 2 3
• 1 is in the new list because it is found in 1 2 3
• 4 is not in the new list because it is not found in 1 2 3
• 1 is in the new list because it is found in 1 2 3
• 5 is not in the new list because it is not found in 1 2 3

Similarly with text, 'Drake Mallard' ∩ 'dark' gives rakaard, because:

• 'D' is not in the new list because it is not found in 'dark'
• 'r' is in the new list because it is found in 'dark'
• 'a' is in the new list because it is found in 'dark'
• 'k' is in the new list because it is found in 'dark'
• 'e' is not in the new list because it is not found in 'dark'
• ' ' is not in the new list because it is not found in 'dark'
• 'M' is not in the new list because it is not found in 'dark'
• 'a' is in the new list because it is found in 'dark'
• 'l' is not in the new list because it is not found in 'dark'
• 'l' is not in the new list because it is not found in 'dark'
• 'a' is in the new list because it is found in 'dark'
• 'r' is in the new list because it is found in 'dark'
• 'd' is in the new list because it is found in 'dark'

Write some APL code that uses ∩ to find the intersection of 'piece of cake' and 'aeiou'. The result should be ieeoae.

In this problem, you’re asked to enter some APL code, including an APL symbol and the data (text or numbers) given to it. Your code will be tested by the APL Challenge system to check whether it gives the correct answer and uses the methods described below.

3: Minimalism

The symbol ⌊ (Lowest) works out which of two numbers is the lowest (the symbol looks like a vertical line with a mark at the lowest point and also looks like the letter L in Lowest). Given a number on each side, it will give the one that has the lowest value. For example, 5 ⌊ 2 gives 2 because 2 is a lower value than 5.

Lowest – and all other basic mathematical symbols of APL – can work on lists of numbers. For example, 5 ⌊ 2 7 1 8 gives 2 5 1 5 because:

• 5 ⌊ 2 gives 2 because 2 is lower than 5
• 5 ⌊ 7 gives 5 because 5 is lower than 7
• 5 ⌊ 1 gives 1 because 1 is lower than 5
• 5 ⌊ 8 gives 5 because 5 is lower than 8

Write some APL code that uses a single ⌊ to limit the numbers in the list 1 2 3 4 5 6 7 8 9 10 so that they are 7 or lower, that is, items with a higher value than 7 are replaced with the value 7. The result should be 1 2 3 4 5 6 7 7 7 7.

Note: In this problem, you’re asked to enter an APL expression (some APL code), including arguments (data given to APL symbols). Your code will be tested by the APL Challenge system to check whether it gives the correct answer and uses the methods described below.

4: Staying In Bounds

Symbols that are used to change data are called functions.

The Highest function ⌈ is just like Lowest from problem 3, except that it gives the higher value of corresponding items (the symbol ⌈ provides a reminder because the mark is at the highest point of the vertical line). For example, 3 1 4 1 5 9 ⌈ 2 7 1 8 2 8 gives 3 7 4 8 5 9, because:

• 3 ⌈ 2 gives 3 because 3 is higher than 2
• 1 ⌈ 7 gives 7 because 7 is higher than 1
• 4 ⌈ 1 gives 4 because 4 is higher than 1
• 1 ⌈ 8 gives 8 because 8 is higher than 1
• 5 ⌈ 2 gives 5 because 5 is higher than 2
• 9 ⌈ 8 gives 9 because 9 is higher than 8

An APL function takes the result of all code to its right as its right argument (the data on the right of the symbol). For example, 2×3+4 gives 14 (rather than the 10 you might expect) because the × takes the result of 3+4 as its right argument. Similarly, -2-6 gives 4 because the left - negates the result of 2-6.

Write an expression that uses ⌊ and ⌈ to limit the numbers in the list 1 2 3 4 5 6 7 8 9 10 so that they are between 3 and 7, that is, items with a lower value than 3 are replaced with the value 3 and items with a higher value than 7 are replaced with the value 7. The result will be 3 3 3 4 5 6 7 7 7 7.

In this problem, you’re asked to enter an APL expression (some APL code), including arguments (data given to APL symbols). Your code will be tested by the APL Challenge system to check whether it gives the correct answer and uses the methods described below.

5: Self Replication

The Copy function ⌿ is used to make copies of items. It takes a list of numbers on its left and any list of equal length on its right. For each number in the list on the left, Copy puts that many copies of the corresponding item of the right argument into the result. For example, 2 0 3 1 2 3 ⌿ 'DYALOG' gives 'DDAAALOOGGG', because:

• 2 copies of 'D' is 'DD'
• 0 copies of 'Y' is ''
• 3 copies of 'A' is 'AAA'
• 1 copies of 'L' is 'L'
• 2 copies of 'O' is 'OO'
• 3 copies of 'G' is 'GGG'

The function ⍳ (iota, a Greek letter) with a number on its right will give you all the whole numbers from 1 to the number that you gave it. (An iota is like an i for integers, which means whole numbers.) For example, ⍳ 7 gives 1 2 3 4 5 6 7

You can use round brackets to change the order in which things are calculated. In problem 4 you learnt that an APL function takes the result of all code to its right as its right argument. This was a simplification; the right argument only goes as far as the nearest closing bracket …) on its right:

• 2×3+4 gives 14 because × takes the result of all code to its right, 3+4, as its right argument.
• 2×(3+4) gives 14 because × takes the result of all code to its right, (3+4), as its right argument.
• (2×3)+4 gives 10 because the right argument of × only goes as far as the nearest closing bracket.

Write an expression that uses ⍳, ⌿, brackets, and the number 5, and returns 1 2 2 3 3 3 4 4 4 4 5 5 5 5 5.

In this problem, you’re asked to enter an APL expression (some APL code), including arguments (data given to APL symbols). Your code will be tested by the APL Challenge system to check whether it gives the correct answer and uses the methods described below.

6: Double Vision

APL has several functions that compare the data on their left with the data on their on the right. They give 1 for yes and 0 for no when answering whether the statement is true. For example: Lower Than (<), Lower Than Or Equal To (≤), Equal To (=), Higher Than Or Equal To (≥), Higher Than (>).

These can compare entire lists. For example, 3 1 4 ≥ 2 1 7 gives 1 1 0 because:

• 1: yes, 3 is bigger than than (or equal to) 2
• 1: yes, 1 is (bigger than than or) equal to 1
• 0: no, 4 is not bigger than than or equal to 7

If you put / (insert) immediately to the right of a function, then it inserts the function between the items of a list on its right. For example, you can use / to multiply all the numbers in a list: ×/ 1 2 3 4 gives 24 because it is the same as 1 × 2 × 3 × 4.

Remember ↓ from problem problem 1? If the number on its left is negative, then it will drop items from the end of the list on its right, rather than from the start.

You can use this to analyse the relationship between adjacent items. For example, let’s say that we have the list 3 1 4 1 5 9 and want to find out which items are higher than their predecessor. To do this:

1. Drop the first and last items (separately), that is, 1 ↓ 3 1 4 1 5 9 and ¯1 ↓ 3 1 4 1 5 9 giving 1 4 1 5 9 and 3 1 4 1 5 respectively.
2. Compare those two lists to find which of the corresponding items has the highest value: 1 4 1 5 9 > 3 1 4 1 5. This gives 0 1 0 1 1.

The two steps put can be together as (1 ↓ 3 1 4 1 5 9) > (¯1 ↓ 3 1 4 1 5 9) giving 0 1 0 1 1.

We can see that the goal has been achieved by placing the result beneath the original list:

3 1 4 1 5 9
 0 1 0 1 1

A 1 in the result indicates that the right item in that pairs of items is greater than the left item.

Write an expression that uses ↓, =, brackets, and + with / to count the number of times that a letter appears immediately after an identical letter in 'bookkeeper'. The result should be 3 ('oo', 'kk', and 'ee').

Note: In this problem, you’re asked to enter your own APL function, without giving it any arguments. Your function will be checked by the APL Challenge system using several tests to see whether it is correct.

7: Double Visions

You can create your own function by putting an expression in curly brackets. For example, the expression 2+3 becomes the function {2+3}. You can let the function take an argument by using ⍵ (omega, the right-most letter of the Greek alphabet) in the expression; any ⍵ is replaced by the argument to the right of the closing curly bracket ( } ).

For example, a function that multiplies its argument by two could be written {2×⍵}. Then:

• {2×⍵} 5 gives 10
• {2×⍵} 7 gives 14

Using what you learnt in problem 6, write a function that takes a word and counts the number of times that a letter appears immediately after an identical letter. When we write APL code and its result together, we often write the code above the result, with six spaces to the left of the code, so for example:

      {solution} 'bookkeeper'
3

      {solution} 'DYALOG'
0

      {solution} 'Abracadabra'
0

      {solution} 'Cryptooology'
2

In this problem, you’re asked to enter your own APL function, without giving it any arguments. Your function will be checked by the APL Challenge system using several tests to see whether it is correct.

8: Unique Characters

The Unique Mask function ≠ takes a list as its right argument and gives a 1 the first time an item appears, 0 for subsequent times. For example, ≠ 3 1 4 1 5 gives 1 1 1 0 1 because:

• 1 because 3 is the first 3
• 1 because 1 is the first 1
• 1 because 4 is the first 4
• 0 because 1 is the second 1
• 1 because 5 is the first 5

In contrast, ≠ 'DYALOG' gives 1 1 1 1 1 1, because every letter only appears once.

The And function ∧ (looks a bit like the letter A for And) gives 1 if both left and right arguments are 1, but 0 if either is 0. For example, 1 1 0 0 ∧ 0 1 0 1 gives 0 1 0 0.

Use ≠ and ∧ combined with / (see problem 6) to write a function that takes a list as its argument and gives 1 if the elements are all unique, 0 otherwise. For example:

      {solution} 'DYALOG APL'
0

      {solution} 1 2 3 4 5 6
1

      {solution} 3 1 4 1 5 9 2 6
0

      {solution} 'copyright'
1

Note: In this problem, you’re asked to enter your own APL function, without giving it any arguments. Your function will be checked by the APL Challenge system using several tests to see whether it is correct.

9: Table Motions

If you put ∘. (jot dot, but sometimes called table) immediately to left of a function, then it produces a table for that function. It does this by inserting the function into each combination that can be made with a number from the left list and a number from the right. If the resulting table had row and column headings, the left argument would be used as the headings for the rows (down the left side) and the right argument would be used as the headings for the columns (across the top).

For example, here’s a multiplication table as you might see it in a book:

And here is the same table in APL form:

      2 3 4 ∘.× 2 3 4 5
4  6  8 10
6  9 12 15
8 12 16 20

In problem 7, you saw how to write a function that takes a right argument using ⍵. If you want to create a function that also takes a left argument, you can use ⍺ (alpha; the left-most letter of the Greek alphabet). For example, a function that sums its left and right arguments and then multiplies by 2 could be written {2×⍺+⍵}.

Write a function that takes lists as arguments and uses ∘. to create an addition table. For example:

      10 20 30 {solution} 4 5 6 7
14 15 16 17
24 25 26 27
34 35 36 37

      3 2 7 2 {solution} 9 9 2 5 9 4
12 12 5  8 12  7
11 11 4  7 11  6
16 16 9 12 16 11
11 11 4  7 11  6

      0 1 0 {solution} 0 1 0 0
0 1 0 0
1 2 1 1
0 1 0 0

In this problem, you’re asked to enter your own APL function, without giving it any arguments. Your function will be checked by the APL Challenge system using several tests to see whether it is correct.

10: X Marks the Spot

The Reverse function ⌽ reverses a list (the symbol looks like a reflection in a mirror). For example, ⌽ 1 2 3 gives 3 2 1.

Remember the And function from problem 8? The Or function ∨ (upside-down And) gives 1 if its left or right argument is 1 (or both are 1), but 0 if both are 0. For example, 1 1 0 0 ∨ 0 1 0 1 gives 1 1 0 1.

Use everything that you’ve learnt so far to write a function that takes a positive integer as its only argument and gives a table with that many rows and columns. The table should have 1s on its diagonals and all other items should be 0. For example:

      {solution} 5
1 0 0 0 1
0 1 0 1 0
0 0 1 0 0
0 1 0 1 0
1 0 0 0 1

      {solution} 4
1 0 0 1
0 1 1 0
0 1 1 0
1 0 0 1

      {solution} 1
1