1089

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1089

The number that was in the envelope all along

Mathematical Curiosity · Magic Tricks · Number Theory · Most-Loved

A magician hands you a sealed envelope and a pencil.

"Think of any three-digit number," she says. "Don't tell me what it is."

You do a little arithmetic, in secret, on a scrap of paper.

Then you open the envelope.

Inside, written in ink before you ever chose your number, is your answer:

1089.

A magician's gloved hand holds a sealed envelope under a spotlight while an audience member hunches over a scrap of arithmetic

Try it on yourself

Here is the trick. You only need a pencil.

1.
Pick a three-digit number whose first and last digits are different - say 532.
2.
Write it backwards: 235.
3.
Subtract the smaller from the larger: 532 − 235 = 297.
4.
Write that answer backwards: 792.
5.
Add them together: 297 + 792 = 1089.

Try another. Start with 731:

731 − 137 = 594, and 594 + 495 = 1089.

Try your own. Try your house number, or the last three digits of your phone number.

You will get 1089 every time.

Step-by-step card trick illustration showing digit tiles for 532 flipping to 235, then subtraction giving 297, then flipping to 792, then addition landing on 1089

One small rule

If your subtraction gives a two-digit answer, keep a zero at the front so it still has three digits. For example, start with 211:

211 − 112 = 099 - reversed, that is 990 - and 099 + 990 = 1089.

(If you forget the zero and reverse 99 as 99, the trick breaks - which is why many magicians ask for a number whose first and last digits differ by at least 2, so the problem never comes up.)

And if the first and last digits are the same - like 525 - the subtraction gives zero, and there is no trick at all.

Uncle Jack's conjuring trick

In 1956, a ten-year-old English boy called David Acheson was mad about magic tricks.

One day he came across an article with a wonderful title: "Abracadabra! Uncle Jack turns you into a Conjuror!"

Inside was a trick with numbers - this one.

He tried it. It worked. He tried it again. It worked again. No matter which number he chose, the answer was always 1089.

He later called it his first mathematical surprise.

The boy never forgot it. He grew up to become a mathematician at Jesus College, Oxford, and in 2002 he wrote a book about the joys of mathematics.

He named it after the trick that had started it all: 1089 and All That.

- A lifelong curiosity, born at a kitchen table.

A 1950s English boy sits cross-legged at a kitchen table, an open magazine beside him showing a cartoon magician, his scrap of paper covered in sums all ending in 1089, his face showing pure astonishment

Why is it always 1089?

It feels like mind-reading. It is actually a squeeze.

Here is the first secret: when you subtract a three-digit number from its own reverse, the middle digits cancel out completely. Only the first and last digits matter.

And the answer is always a multiple of 99:

99 × (first digit − last digit)

So whatever number you start with, step 3 can only ever give you one of a tiny handful of answers:

099 198 297 396 495 594 693 792 891

Look at them closely. In every one, the middle digit is 9, and the first and last digits add up to 9.

Now watch what happens when you add each one to its reverse:

099 + 990 = 1089

198 + 891 = 1089

297 + 792 = 1089

396 + 693 = 1089

495 + 594 = 1089

… and the rest do the same.

The ones column always adds to 9. The tens column always adds to 18. The hundreds column always adds to 9, plus the 1 carried over.

Every road leads to the same four digits.

Hundreds of possible starting numbers get squeezed into just nine possibilities - and those nine all collapse into one.

A whimsical sorting machine: hundreds of three-digit number cards pour in at the top, narrow into nine labelled slots, pair with their reverses, and all collapse into a single glowing 1089 at the bottom

A familiar face

In the step-3 list sits 495 - and 495 appears on the /6174 page too. It is where Kaprekar's "biggest-minus-smallest" rule traps every three-digit number. The two tricks are close cousins: both work by subtracting a number from a rearrangement of its own digits.

1089 in the mirror

The trick is not the only secret 1089 keeps.

Multiply it by 9:

1089 × 9 = 9801

The answer is 1089 written backwards.

Now watch its whole times table:

1089 × 1 = 1089

← → 9801 = 1089 × 9

1089 × 2 = 2178

← → 8712 = 1089 × 8

1089 × 3 = 3267

← → 7623 = 1089 × 7

1089 × 4 = 4356

← → 6534 = 1089 × 6

1089 × 5 = 5445

← its own reflection

1089 × 6 = 6534

1089 × 7 = 7623

1089 × 8 = 8712

1089 × 9 = 9801

Read it from the top and from the bottom at the same time. Each answer is the mirror image of its partner: 1089 and 9801, 2178 and 8712, 3267 and 7623, 4356 and 6534. And 5445, right in the middle, is its own reflection.

There is even a hidden bonus: 2178 × 4 = 8712 - another number that, when multiplied, turns itself backwards.

And one more: 1089 is 33 × 33, while its mirror, 9801, is 99 × 99.

A symmetrical mirror illustration showing the 1089 times table with each answer reflected as its digit-reversed partner, and 5445 sitting exactly on the mirror line

Hardy's "odd fact"

The mirror trick caught the eye of one of the most famous mathematicians of the 20th century: G. H. Hardy - the same man who, in the story told on the /1729 page, arrived at Ramanujan's hospital bedside in a taxi numbered 1729.

In 1940, Hardy wrote a short book called A Mathematician's Apology, about what makes mathematics beautiful.

In it, he pointed out that 8712 and 9801 are the only four-figure numbers that are exact multiples of their own reversals:

8712 = 4 × 2178 9801 = 9 × 1089

Then he dismissed them. He called such things "odd facts, very suitable for puzzle columns and likely to amuse amateurs," with nothing in them that appeals much to a mathematician.

Hardy was a very great mathematician. But here, perhaps, he was a little unfair: since then, mathematicians have written more than a dozen research papers exploring exactly these reversal puzzles.

G. H. Hardy in a 1940s study, fountain pen raised, a slightly dismissive eyebrow, 9801 equals 9 times 1089 written on the page before him, a vintage taxi with 1729 visible through the window behind

The magician's favourite number

Because the answer never changes, 1089 is a gift to magicians.

They can write it inside a sealed envelope. Hide it on a playing card. Or ask you to open a book at page 108, count down to line 9 - and read out a word they "predicted" hours ago.

The audience thinks the magician has read their mind.

What really happened is simpler, and in a way more wonderful: the audience's own arithmetic did the magic. However freely you chose your number, the rules quietly steered you to the same place.

A magician on a small stage reveals a card marked 1089 while an astonished audience clutches scraps of paper covered in arithmetic

Freedom and fate

The 1089 trick is a tiny lesson about something big.

It feels as if you had complete freedom. You could have chosen your birthday, your house number, anything at all.

But hidden inside the rules was a structure that did not care what you chose. Every path was already bending toward the same answer.

Mathematicians love tricks like this because they turn a feeling of magic into a feeling of understanding. Once you see the 99s and the 9s, the mystery does not disappear - it moves. It stops being "how did she know?" and becomes "how wonderful that numbers work like this at all."

Many small paper boats set off from different points on a wide river, hidden currents guiding them all to a harbour gate marked 1089

1089 in a nutshell

Take a three-digit number whose first and last digits are different. Reverse it and subtract the smaller from the larger. Reverse the answer and add. You always get 1089 - as long as you keep a leading zero, as in 099.

The subtraction always gives a multiple of 99, with a 9 in the middle, and every one of those adds to its reverse to make 1089.

A ten-year-old David Acheson met the trick in 1956 and later named a book after it: 1089 and All That.

1089 × 9 = 9801, its own reflection.

G. H. Hardy - of taxi-cab 1729 fame - called this kind of thing an "odd fact."

Magicians have loved it ever since.

A recap panel combining the sealed 1089 envelope, the boy with the magazine, the sorting machine, the mirror with 1089 and 9801, and Hardy at his desk with the 1729 taxi

Try it on yourself

Uncle Jack's conjuring trick

Why is it always 1089?

Enjoying this story?

1089 in the mirror

Hardy's "odd fact"

The magician's favourite number

Freedom and fate

1089 in a nutshell

Other numbers with stories worth telling

See a strange number.Become curious.Discover a story.

See a strange number.
Become curious.
Discover a story.