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Clock

The angle between the hands at 3:20, and the time 3 h 50 min after 10:45 pm — across noon and midnight, 12- or 24-hour, finer minutes, longer spans.

  • Calculation
  • Timed
  • Endless levels

How it plays

At h:mm the minute hand is at 6° a minute past 12; the hour hand at 30° an hour plus half a degree a minute — it does not wait on the hour. The angle asked is the smaller of the two between them, so never more than 180°.

The time after a duration: add the minutes, carry into the hours, and wrap at midnight — on a 12-hour clock, at 12, flipping am and pm. Durations are sometimes given as minutes alone, which must be turned into hours first.

Levels move from the half hour to any minute and half-minutes, lengthen the durations from minutes to days without end, and shorten the time. The usual slips — the hour hand left on the hour, an hour lost in the carry, am and pm swapped — are among the options.

Keys

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

What it trains

This trains clock arithmetic and the hand-angle calculation. It is practice at these, nothing wider.

Learn

How to get better at clock: 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. Read which question it is: the angle between the hands at a time, or the time after a duration. The prompt shows “h:mm” alone for an angle and “time + duration” for the other.
  2. For an angle, put both hands in degrees from 12: the minute hand at 6° a minute, the hour hand at 30° an hour plus 0.5° a minute. Take the difference.
  3. If the difference is more than 180°, take it from 360°. The question always wants the smaller angle.
  4. For a duration, turn it into hours and minutes first, drop whole days, add the hours, then the minutes, and wrap past midnight (or past 12 on the 12-hour clock, flipping am and pm).

Techniques and tricks

  • The one-line angle formula

    The angle is |30 × hours − 5.5 × minutes|. The 5.5 is the minute hand’s 6° less the hour hand’s 0.5°, because both move.

    e.g. 3:20: 30 × 3 − 5.5 × 20 = 90 − 110 = −20, so 20°.

  • Halve the minutes for the hour hand

    The hour hand creeps half a degree a minute, so it sits “minutes ÷ 2” degrees past its hour mark. At 9:45 it is 22.5° past the 9.

    e.g. 9:45: hour hand 270 + 22.5 = 292.5°, minute hand 270°, angle 22.5°.

  • Reflex check

    When the formula gives more than 180°, subtract it from 360°. Hands far apart on the face are close the other way round.

    e.g. 10:10: 300 − 55 = 245°, and 360 − 245 = 115°.

  • Half-minutes as 5.5 × the half

    With seconds of :30, use the minutes as m.5. The hour hand adds a quarter of a degree for the half-minute and the minute hand adds 3°.

    e.g. 4:20:30: 30 × 4 − 5.5 × 20.5 = 120 − 112.75 = 7.25°.

  • Minutes into hours by sixties

    For a duration given in minutes alone, take off 60s until under 60. Hundreds are not hours: 527 minutes is not 5 h 27 min.

    e.g. 527 min: 8 × 60 = 480, 47 left, so 8 h 47 min.

  • Fill to the hour first

    Add just enough minutes to reach the next whole hour, then add the hours, then whatever minutes are left. It avoids carrying a 60.

    e.g. 07:47 + 7 h 13 min: 13 min takes you to 08:00, then 7 h makes 15:00.

  • Drop whole days

    Every 24 hours (1,440 minutes) brings the clock back where it started. Remove them before adding anything else.

    e.g. 1 day 2 h 22 min after 19:50 is just 2 h 22 min after it: 22:12.

As the levels climb

  • The times get finer: on the half hour, then the quarter, every 5 minutes, any minute, and finally half-minutes, reached by level 10. Each step adds base time; meet it by switching from picturing the face to using 30h − 5.5m.
  • Durations lengthen without end: up to 1 h 30 min at first, then about 4½, 8¾, 14 and 20 hours, and past a day from level 12. Past a day, drop the whole days before anything else.
  • The clock closes to 85%, 72% and then 62% of the base time, reached by level 9. Fill-to-the-hour and the formula are the two moves fast enough to survive it.
  • Minute-only durations (“in 527 minutes”) come up about three times in ten and earn 3 extra seconds. Spend them on the conversion, which is where the mistakes are.

Traps to avoid

  • The hour hand left on its hour mark. At 3:20 it gives 30°, not 20°. If the minutes are not zero, the hour hand has moved half a degree for each of them.
  • The reflex angle offered as the answer: 340° for 3:20. Anything over 180° is the long way round.
  • A few degrees off either way, from using the full minutes instead of half of them for the hour hand, or 5° out from a slip in the 5.5 times table. Check the half-minutes part on its own.
  • An hour out, either way, from carrying 60 minutes wrongly; or 12 hours out, which on the 12-hour clock is am and pm swapped. Say “past midnight” or “past noon” aloud when the sum crosses one.
  • Minutes alone read as hours and minutes: 527 minutes taken as 5 h 27 min. Always divide by 60.

How to practise

  • Use the Questions setting to practise “Angle between the hands” and “Time after a duration” separately; the methods share nothing.
  • Set Clock to “12-hour, am and pm” once the 24-hour clock is easy. The 12-hour wrap is where most time answers go wrong.
  • Switch Answer by to “Typing the answer” when the options feel easy: options let you spot the reflex and frozen-hand lures without finishing the sum.
  • Learn 5.5 times the minutes for the common minutes (5.5 × 10 = 55, × 20 = 110, × 30 = 165, × 40 = 220, × 50 = 275) so the formula costs one subtraction.

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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