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Off by one

Fence posts, cuts, floors, inclusive pages, a striking clock, lists numbered from zero — count the ends right, every time, at speed.

  • Logic
  • Timed
  • Endless levels

How it plays

Each question is a counting situation where the first number that comes to mind is one out: a 100 m fence with a post every 10 m has 11 posts, not 10; cutting a log into 5 pieces takes 4 cuts.

New situations unlock as you climb — closed loops, days inclusive, clocks that strike, doses on a schedule, indexes from zero, both sides of a road, pages of results — and the numbers grow without end.

The off-by-one answer is always among the options. The working names which end was missed.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This trains the habit of counting the ends — posts against gaps — in the situations it shows. It claims nothing wider.

Learn

How to get better at off by one: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Decide what you are counting: the things (posts, pages, tablets, strikes) or the spaces between them (gaps, flights, cuts, seconds). Every question here turns on that one choice.
  2. Work out the easy number first — usually a subtraction or a division, which counts the spaces.
  3. Then ask about the ends. In a straight line with both ends included, things are one more than spaces. In a closed loop they are equal. With both ends left out, things are one fewer.
  4. If you are unsure, shrink the question to a tiny case you can picture — a fence with two posts, pages 3 to 4 — and see which way the rule goes, then apply it to the real numbers.

Techniques and tricks

  • Posts are gaps plus one

    On a straight line with something at both ends, divide the length by the spacing to get the gaps, then add one.

    e.g. A 60 m fence with a post every 5 m: 60 ÷ 5 = 12 gaps, so 13 posts

  • A loop has no spare post

    Round a closed shape, the last gap ends at the first post, so posts equal gaps. Divide and stop.

    e.g. 120 m round a field, a post every 10 m: 12 posts

  • Inclusive is minus, then plus one

    From a to b with both included is b − a + 1. Strictly between, with both left out, is b − a − 1.

    e.g. Pages 14 to 37: 37 − 14 + 1 = 24. Whole numbers strictly between 14 and 37: 22

  • Count the gaps, not the strikes

    For anything timed between events — a clock striking, tablets on a schedule — the time belongs to the gaps. Find the time of one gap, then multiply by the gaps you need.

    e.g. 6 strikes in 15 s is 5 gaps of 3 s, so 9 strikes take 8 × 3 = 24 s — not 22.5, not 27

  • Floors are posts, flights are gaps

    Going from floor a to floor b takes b − a flights; you do not climb a flight to reach the floor you start on. The same goes for cuts: pieces − 1.

    e.g. Floor 3 to floor 11 is 8 flights; 7 pieces of log need 6 cuts

  • Round up for a part page

    When things are shared out in fixed batches, a part-filled last batch still counts. Divide and round up, never down.

    e.g. 245 results at 20 a page: 245 ÷ 20 = 12.25, so 13 pages

  • Multiples are steps plus one

    When both ends are multiples of k, the count is (high − low) ÷ k + 1 — the same posts-and-gaps rule, with k as the spacing.

    e.g. Multiples of 7 from 21 to 84: (84 − 21) ÷ 7 = 9, so 10

As the levels climb

  • The ladder starts with fence posts, cuts and pages inclusive. The next level adds floors and stairs, strictly between, closed loops and days inclusive; then the numbers grow and the clock closes to 80%.
  • Next come the striking clock, doses on a schedule and indexes from zero, then bigger numbers and the clock at 65%. The last set is both sides of the road, multiples in a range and pages of results, and the clock settles at 55%.
  • From there only the numbers grow, without end. A newly opened kind makes up a larger share of the questions for a while, so expect to meet it often just after it arrives.
  • The clock starts at 9 seconds and gains 1.2 for each step the numbers grow, then is scaled down by the level’s share; typing gets 20% more. Bigger numbers do not change the rule, so decide posts or gaps first and the arithmetic is quick.

Traps to avoid

  • The plain division. 100 m with a post every 10 m is 10 gaps — but 11 posts. That answer, one short, is always among the options.
  • One too many in a loop. Adding one round a field counts the first post twice; a closed fence has as many posts as gaps.
  • Scaling the clock by strikes. 15 s for 6 strikes is not 2.5 s a strike; the time is in the 5 gaps. For 9 strikes the lures are 23 s (22.5 rounded) and 9 × 3 = 27 s.
  • Counting floors instead of flights, or dropping the first day of a festival. Say aloud whether the start itself is counted: you stand on the first floor, but you live through the first day.
  • Doing only one side of the road, or doubling before adding the end tree. Work out one side fully — gaps plus one — then double it.

How to practise

  • Switch “Answer by” to “Typing the number” once the options feel easy. The off-by-one answer is always offered, so choosing can let you spot it; typing makes you count the ends yourself.
  • Lower “Options shown” to 2 for a sharp test: the right count against its most tempting slip, with nothing else to rule out.
  • Before each answer, say one word: “posts” or “gaps”. It takes a second and it is the whole skill here.
  • When you miss one, read which end the working says you missed, then picture the two-post version of that same question.

Levels

Each question has a clock that comes from the work it asks for. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

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