Numbers and digits are the same

Always the same result

"You have to explain that to me!" There was indignation in Maria's voice as she thundered her school backpack on the table and collapsed on a chair, exhausted.

I was completely unclear what she had in mind. But I didn't need to ask, because sooner or later she would say what was going on.

But at first she said nothing. She fished a pen out of her backpack, pulled the newspaper over to her, wrote the number 853, including 358, on the margin, subtracted the numbers from each other, got 495 as the result, underlined this number and then said: “That always comes out! "

"But Maria," I had to slow her down, "if you subtract any number from each other, any number can come out as a result."

"You don't understand anything!" She was right, in fact I didn't understand what she was trying to say to me.

I tried again: “How did you choose 853? Could you have chosen 537? "

"Of course not. You have to take the largest number, well, "she took up the pen and energetically wrote 753. And while she was already explaining something to me, she said:" Now you write the smallest number below it, 357. And you have to subtract it . "

"Aha, so you create the largest and smallest number from three digits and subtract the smaller one from the larger one."

She didn't have that, but I took care not to contradict her.

"And what should come out of there?"

"Always the same", was her short answer, "495."

I did the math: 753 minus 357 equals 396. “But this doesn't come out as 495,” I dared to say.

"You have to do that again now, of course!" Apparently I was too dumb to her today. She wrote and spoke the numbers slowly and clearly: 963 minus 369 is 594. "You see, and now again: 954 minus 459 is 495. So that's right!"

"So you take the result that you get when you pull it off, build the largest and smallest number out of it and ..."

I was not allowed to finish. "Yes. And you always come to 495. And I've been telling you for half an hour that I want to know why that is! "

I slowly felt my way: “The two numbers made up of three digits, the largest and the smallest, have the same digit in the middle. And with the ones digit, a larger number is deducted from a smaller one. "

Maria understood: "That's why you have to keep 1, and then again in the middle."

“Yes, and that means the middle digit of the result is always 9.” Maria checked that. “Besides,” I continued, “the difference is a number that is always divisible by 9. Therefore the cross sum is a nine, and therefore the first and last digits add up to 9. "

Maria was amazed. "In my numbers, the first digit is 4 and the last 5, and your first digit is 3 and the last 6, so together also 9."

I continued: “So only very few numbers come into question as differences. Can the first digit be a 1? "

Maria considered: "Then the last one has to be an 8, the middle one is 9 anyway, so 198."

“Can the first digit be a 2?” Maria didn't appreciate my answer, but wrote: 198, 297, 396, 495, 594, 693, 792, 891.

"Now you just have to try out whether it works," she said.

“Or you can think of something. Let's look at the last number, 891. We rearrange it to 981 or 189 and subtract the numbers from each other. That gives 792. You're not done yet, but you have calculated a number that is still in the middle of the row. "

Maria understood: "And if you do that with this number, you come back a little further in the middle - until you are finally at 495."

“You can understand this, magical property‘ quite well, ”I said. Maria didn't answer, but she seemed satisfied.

September 20, 2011

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