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Higher or lower

Find a hidden number from what each guess says: higher or lower. Halve what is left each time and you never need more than log₂ guesses.

  • Logic
  • Untimed
  • Endless levels

How it plays

A number is hidden somewhere from 1 to N. Pick one of the guesses offered; the answer is higher, lower, or yes. The range left is shown after every guess.

The middle of what is left is always among the offers. Guessing it halves the range every time, which finds any number in ⌈log₂(N+1)⌉ guesses — that is the par. The range grows with the level, without end.

Keys

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

What it trains

This practises binary search — halving an interval and finding its midpoint in the head. It claims nothing beyond that.

Learn

How to get better at higher or lower: 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 the range left, low to high. It is shown after every guess, so you never need to remember the earlier answers.
  2. Work out the middle: add the two ends and halve, rounding down. That guess leaves at most half of the range, whichever way the answer goes.
  3. Find that exact number among the offers and pick it. The middle is always offered, so if your number is not there, check your sum.
  4. After the answer, the new range is the half on the side it pointed to, not counting the number you guessed. Repeat until the answer is yes.

Techniques and tricks

  • Add the ends and halve

    The middle of low to high is (low + high) ÷ 2, rounded down. Halve the sum, not just the top number.

    e.g. 300 to 1,000: 300 + 1,000 = 1,300, so 650

  • Low plus half the gap

    With big numbers, find the width, halve it, and add it to the low end. It keeps the numbers smaller in your head.

    e.g. 4,001 to 5,000: the gap is 999, half is 499, so 4,001 + 499 = 4,500

  • Halve an odd sum

    When low + high is odd, halve the even number below it. Dropping the half keeps you on a whole number the game offers.

    e.g. 51 to 100: 151 is odd, so halve 150 to get 75

  • Know your doubling

    k guesses can settle 2^k − 1 numbers: 1, 3, 7, 15, 31, 63, 127, 255, 511, 1,023. That is where the par comes from.

    e.g. 1 to 1,000 has par 10, because 511 is too few and 1,023 is enough

  • Check the budget

    With r guesses left, the range must be no wider than 2^r − 1. If it is wider after an answer, you guessed off the middle earlier — find where.

    e.g. 3 guesses left and 9 numbers in the range: 9 is more than 7, so the worst case now goes over par

As the levels climb

  • Each level widens the range in steps of 1, 2 and 5: 1 to 10, 20, 50, 100, 200, 500, 1,000, and on up to a million.
  • Past a million the top grows by a million a level, without end. The par goes up by one each time the top passes one less than a power of two — 1,048,575, then 2,097,151, then 4,194,303 — so it climbs slowly: 20 guesses for a million, 21 for two million, 22 for three and four.
  • Bigger ranges mean bigger sums to halve, often six or seven digits. Switch to Low plus half the gap once the numbers pass a few thousand.
  • There is no clock. The only limit is the par, and going over it still earns part credit — par divided by the guesses you took.

Traps to avoid

  • Halving the top number instead of the range. For 300 to 1,000, half of 1,000 is 500 — and 500 is offered alongside the real middle, 650.
  • Keeping the guessed number in the range. After “higher than 75”, the range starts at 76, not 75. Including it shifts every later middle.
  • Guessing near the middle and calling it close enough. For 1 to 1,000 with par 10, a first guess anywhere from 489 to 512 still works, but 480 leaves 520 numbers above it — more than the 511 that nine guesses can settle.
  • Picking the button in the middle of the row. With “Offers shown” on “Shuffled”, and often even in order, the middle offer on screen is not the middle of the range.

How to practise

  • Set “Offers shown” to “Shuffled” once you can find the middle quickly. It stops you leaning on where a number sits in the row.
  • Raise “Guesses offered” to 8 to put more numbers on screen. The more there are, the more an estimate lands on the wrong one and only the exact sum will do.
  • Say the sum and the half aloud — “one thousand three hundred, six fifty” — before you look at the offers.
  • After each game, read the line of middles in the working and compare it guess for guess with yours.

Levels

It is untimed unless you turn the clock on. 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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