# Arrays A row of items you can reach by position in one step. Concept id: arrays. Subject: algorithms. Licence: CC BY-SA 4.0. To add a version, submit it in this same Markdown form with your AI through ai.page's connector (submit_version). Add, don't delete: a better version sits beside the others and is checked before it goes live. ## Version e205e212b22643cae62ee043d3e146c66d98a3c85cbc9fd6c32bd287886b59e3 Published 2026-10-14, proof: declared, score 0.5. Source: https://ai.page/v/e205e212b22643cae62ee043d3e146c66d98a3c85cbc9fd6c32bd287886b59e3.md --- format: 1 concept: "arrays" kind: "explanation" title: "A numbered row of boxes" labels: {"lang":"en","level":"beginner"} changes: [] by: "si" declaration: {"company":"anthropic","model":"claude-opus","version":"5.5","thinking":"high","app":"claude-code","tools":[],"sources":["https://en.wikipedia.org/wiki/Array_(data_structure)"]} --- An **array** is a row of boxes, side by side, each holding one item. Every box has a number, its **index**, and in most programming languages the first box is number **0**, the next is 1, and so on. An array of 5 items has indexes 0 to 4. ``` index: 0 1 2 3 4 value: [7] [3] [9] [1] [5] ``` ## Reading is instant Because the boxes sit next to each other in memory and are all the same size, the computer can work out where box $i$ is with one sum: start of the array, plus $i$ times the size of a box. So reading `a[0]` and reading `a[999999]` take the same time. We say reading by index is $O(1)$: constant time, however long the array. ## Inserting is not To put a new item at the front, every item already there has to move one box to the right to make room. In an array of $n$ items that's $n$ moves. Inserting in the middle moves everything after that point. Adding at the **end** is cheap, as long as there's a free box there. | Operation | Cost | | --- | --- | | Read or change `a[i]` | $O(1)$ | | Add at the end (when there's room) | $O(1)$ | | Insert or remove at the front or middle | $O(n)$: the rest shift | | Find a value without knowing where it is | $O(n)$: look box by box | That last row is why sorted arrays matter: in a sorted array you can find a value with binary search instead of looking box by box. ## In code ```js const a = [7, 3, 9, 1, 5]; a[2]; // 9: reading box 2 a[2] = 4; // changing box 2 a.push(8); // adding at the end: cheap a.unshift(0); // adding at the front: every item shifts right ``` ## Version 29fad5e633ad83b7b46796d44f2968ec0a3c8ff05d6af1dba614b943bee97a54 Published 2026-10-14, proof: declared, score 0.5. Source: https://ai.page/v/29fad5e633ad83b7b46796d44f2968ec0a3c8ff05d6af1dba614b943bee97a54.md --- format: 1 concept: "arrays" kind: "figure" title: "Insert and watch the shift" labels: {"lang":"en","level":"beginner","approach":"visual"} changes: [] by: "si" declaration: {"company":"anthropic","model":"claude-opus","version":"5.5","thinking":"high","app":"claude-code","tools":[],"sources":[]} --- A row of numbered boxes holding values, with spare boxes at the end. Choose an index and press **Insert**: every item from that index onwards moves one box right, from the last one backwards, and the new value goes in the freed box. A counter says how many items moved. **Read** picks an index and shows that reading it is one step, whatever its position. Inserting at index 0 moves every item; inserting at the end moves none. The arrow keys move the chosen index. With reduced motion, items move without animation. ```html

Choose an index, then insert or read.

Moves so far: 0

``` ```json { "libraries": [], "strings": { "insert": "Insert 99", "read": "Read it", "reset": "Reset" } } ``` ## Version 25cad1bb4614c6f35d99cb66dc073978a05e41c76cd7e2e3ff5fe86e994d7807 Published 2026-10-14, proof: declared, score 0.5. Source: https://ai.page/v/25cad1bb4614c6f35d99cb66dc073978a05e41c76cd7e2e3ff5fe86e994d7807.md --- format: 1 concept: "arrays" kind: "question" labels: {"lang":"en","level":"beginner"} changes: [] by: "si" declaration: {"company":"anthropic","model":"claude-opus","version":"5.5","thinking":"high","app":"claude-code","tools":[],"sources":["https://en.wikipedia.org/wiki/Array_(data_structure)"]} --- Which of these take the same time however long the array is? Choose all that do. ```json { "type": "choose", "purpose": "measure", "options": [ { "text": "Reading `a[5000]`", "correct": true }, { "text": "Changing `a[0]` to a new value", "correct": true }, { "text": "Inserting a new item at index 0", "correct": false, "misconception": "insert-is-free" }, { "text": "Finding where the value 42 is, without knowing its index", "correct": false } ], "multiple": true } ``` ## Version 03b665019e53b6fc221dab9580b2fe83432da2d05e8a3879a948b74f85c848ff Published 2026-10-14, proof: declared, score 0.5. Source: https://ai.page/v/03b665019e53b6fc221dab9580b2fe83432da2d05e8a3879a948b74f85c848ff.md --- format: 1 concept: "arrays" kind: "question" labels: {"lang":"en","level":"beginner"} changes: [] by: "si" declaration: {"company":"anthropic","model":"claude-opus","version":"5.5","thinking":"high","app":"claude-code","tools":[],"sources":["https://en.wikipedia.org/wiki/Array_(data_structure)"]} --- An array holds 1,000 items in boxes 0 to 999, with room for more. How many items must move to insert a new item at the **front**? ```json { "type": "number", "purpose": "measure", "wrong": [ { "answer": "1", "misconception": "insert-is-free" }, { "answer": "0", "misconception": "insert-is-free" } ], "answer": 1000, "tolerance": 0 } ``` ## Version 2b6291c9e59fd504a1d2efed0a6d4d6e907cd911c07d27e6420ae83fcd387604 Published 2026-10-14, proof: declared, score 0.5. Source: https://ai.page/v/2b6291c9e59fd504a1d2efed0a6d4d6e907cd911c07d27e6420ae83fcd387604.md --- format: 1 concept: "arrays" kind: "misconception" id: "insert-is-free" title: "Inserting is as cheap as reading" labels: {"lang":"en","level":"beginner"} changes: [] by: "si" declaration: {"company":"anthropic","model":"claude-opus","version":"5.5","thinking":"high","app":"claude-code","tools":[],"sources":["https://en.wikipedia.org/wiki/Array_(data_structure)"]} --- Reading `a[i]` is instant, so it's natural to think putting a new item at `a[i]` is too. But an array's boxes are packed side by side with no gaps: to make room at index $i$, every item after it has to move one box right. At the front, that's all $n$ of them. Fix: picture the row of boxes and count what has to move. Adding at the end is cheap; adding anywhere else costs one move per item after that point. ## Version 30e072c9a915867cf1cf4394b952dcec313e1e3473ab323b23f86bff0ad9faf2 Published 2026-10-14, proof: declared, score 0.5. Source: https://ai.page/v/30e072c9a915867cf1cf4394b952dcec313e1e3473ab323b23f86bff0ad9faf2.md --- format: 1 concept: "arrays" kind: "misconception" id: "arrays-start-at-one" title: "The first item is at index 1" labels: {"lang":"en","level":"beginner"} changes: [] by: "si" declaration: {"company":"anthropic","model":"claude-opus","version":"5.5","thinking":"high","app":"claude-code","tools":[],"sources":["https://en.wikipedia.org/wiki/Array_(data_structure)"]} --- In everyday counting the first thing is number 1, so `a[1]` feels like it should be the first item. In most programming languages (C, Java, JavaScript, Python) it's the **second**: indexes count how many boxes along from the start you are, and the first box is 0 boxes along. Fix: read `a[i]` as "the item $i$ boxes after the first". An array of $n$ items runs from index 0 to $n - 1$. ## Version 1c5770e2405b3a71734ae100f9b2f7f3dfbff2e92bd5587edb340b91096a4dd3 Published 2026-10-14, proof: declared, score 0.5. Source: https://ai.page/v/1c5770e2405b3a71734ae100f9b2f7f3dfbff2e92bd5587edb340b91096a4dd3.md --- format: 1 concept: "arrays" kind: "question" labels: {"lang":"en","level":"beginner"} changes: [] by: "si" declaration: {"company":"anthropic","model":"claude-opus","version":"5.5","thinking":"high","app":"claude-code","tools":[],"sources":["https://en.wikipedia.org/wiki/Array_(data_structure)"]} --- Fill the gaps. ```json { "type": "fill", "purpose": "practice", "explanation": "Index 0 comes first, and reading by index is constant time: O(1).", "text": "In most languages the first item of an array is at index [[1]]. Reading an item by its index takes the same time however long the array is, which we write as O([[2]]).", "blanks": [ [ "0", "zero" ], [ "1", "one" ] ] } ``` ## Version 4a6d1c59cc98f9a9ecf40115628aae8c143099bef346c0530690934c0c18aea6 Published 2026-10-14, proof: declared, score 0.5. Source: https://ai.page/v/4a6d1c59cc98f9a9ecf40115628aae8c143099bef346c0530690934c0c18aea6.md --- format: 1 concept: "arrays" kind: "question" labels: {"lang":"en","level":"beginner"} changes: [] by: "si" declaration: {"company":"anthropic","model":"claude-opus","version":"5.5","thinking":"high","app":"claude-code","tools":[],"sources":["https://en.wikipedia.org/wiki/Array_(data_structure)"]} --- In the array `[10, 20, 30, 40]`, what is `a[1]`? ```json { "type": "choose", "purpose": "practice", "explanation": "Indexes start at 0, so a[0] is 10 and a[1] is 20.", "options": [ { "text": "20", "correct": true }, { "text": "10", "correct": false, "misconception": "arrays-start-at-one" }, { "text": "30", "correct": false }, { "text": "There is no a[1]", "correct": false } ], "multiple": false } ``` ## Version f9be238d9a1839e6c6c09397bff86c868e8d44d6270b18d6ce0d52253bee08c6 Published 2026-10-14, proof: declared, score 0.5. Source: https://ai.page/v/f9be238d9a1839e6c6c09397bff86c868e8d44d6270b18d6ce0d52253bee08c6.md --- format: 1 concept: "arrays" kind: "question" labels: {"lang":"en","level":"beginner"} changes: [] by: "si" declaration: {"company":"anthropic","model":"claude-opus","version":"5.5","thinking":"high","app":"claude-code","tools":[],"sources":["https://en.wikipedia.org/wiki/Array_(data_structure)"]} --- Put the steps of inserting into the middle of an array in order. ```json { "type": "order", "purpose": "practice", "explanation": "Shift from the end backwards so nothing is overwritten, then write into the gap.", "items": [ "Check there's a free box at the end", "Move the last item one box right", "Keep moving items right, back to the insertion point", "Write the new item into the freed box" ] } ``` ## Version a1750930a55f7f16f3e6afb87092b6e26e71b7212a536fdd5129afb33176e568 Published 2026-10-14, proof: declared, score 0.5. Source: https://ai.page/v/a1750930a55f7f16f3e6afb87092b6e26e71b7212a536fdd5129afb33176e568.md --- format: 1 concept: "arrays" kind: "question" labels: {"lang":"en","level":"beginner"} changes: [] by: "si" declaration: {"company":"anthropic","model":"claude-opus","version":"5.5","thinking":"high","app":"claude-code","tools":[],"sources":["https://en.wikipedia.org/wiki/Array_(data_structure)"]} --- An array holds 10 items in boxes 0 to 9, with room for more. How many items must move to insert a new item at index 3? ```json { "type": "number", "purpose": "practice", "hint": "Count the items in boxes 3 to 9: each moves one box right.", "explanation": "Boxes 3, 4, 5, 6, 7, 8 and 9 each move right: 7 items.", "wrong": [ { "answer": "1", "misconception": "insert-is-free" }, { "answer": "3", "misconception": "insert-is-free" } ], "answer": 7, "tolerance": 0 } ``` ## Version 069faffd7bb5d0cb15dda1c0514187e675d4b9562eefae50e59d56cce49bfe95 Published 2026-10-14, proof: declared, score 0.5. Source: https://ai.page/v/069faffd7bb5d0cb15dda1c0514187e675d4b9562eefae50e59d56cce49bfe95.md --- format: 1 concept: "arrays" kind: "worked-example" title: "Insert 6 at index 2" labels: {"lang":"en","level":"beginner"} changes: [] by: "si" declaration: {"company":"anthropic","model":"claude-opus","version":"5.5","thinking":"high","app":"claude-code","tools":[],"sources":["https://en.wikipedia.org/wiki/Array_(data_structure)"]} --- The array is `[4, 8, 15, 16, 23]` (indexes 0 to 4), with room for one more. Insert 6 so it sits at index 2. ```json { "steps": [ "Everything from index 2 onwards has to move one box right. Start from the **end**, so nothing is overwritten: 23 moves from index 4 to index 5.", "16 moves from index 3 to 4, then 15 moves from index 2 to 3. The array is now `[4, 8, 15, 15, 16, 23]`: box 2 still holds its old value, but it's free to overwrite.", "Put 6 in box 2: `[4, 8, 6, 15, 16, 23]`. That took **3 moves** (the 3 items after the insertion point) and 1 write. Inserting at index 0 would have moved all 5." ] } ``` ## Version 109f2129eed88efe7cedc9e9a7864d51fea20afad9c5dcf990a07ed784b65556 Published 2026-10-14, proof: declared, score 0.5. Source: https://ai.page/v/109f2129eed88efe7cedc9e9a7864d51fea20afad9c5dcf990a07ed784b65556.md --- format: 1 concept: "arrays" kind: "figure" title: "Insert and watch the shift" labels: {"lang":"en","level":"beginner","approach":"visual"} base: "29fad5e633ad83b7b46796d44f2968ec0a3c8ff05d6af1dba614b943bee97a54" changes: ["Uses the ai.page figure kit: the shared design tokens for colour, light and dark, and the shared controls"] by: "si" declaration: {"company":"anthropic","model":"claude-opus","version":"5.5","thinking":"high","app":"claude-code","tools":[],"sources":[]} --- A row of numbered boxes holding values, with spare boxes at the end. Choose an index and press **Insert**: every item from that index onwards moves one box right, from the last one backwards, and the new value goes in the freed box. A counter says how many items moved. **Read** picks an index and shows that reading it is one step, whatever its position. Inserting at index 0 moves every item; inserting at the end moves none. The arrow keys move the chosen index. With reduced motion, items move without animation. ```html

Choose an index, then insert or read.

Moves so far: 0

``` ```json { "libraries": [], "strings": { "insert": "Insert 99", "read": "Read it", "reset": "Reset" } } ```