# Observable mathematical phenomena

• Mathematica
Lets share some mathematical phenomena that are just cool (for math people)

if you multiply any number of integers whose digits are only 1, the product will always be palandromic. Grab a calculator and see for your self. I would be really interested to see a proof for this phenomena.

heres another one, you;ve probably heard of it before.
take a 4 digit integer. All are > 0, and a maximum of 3 of the digits can be equal.

rearrange the digits in such a fashion where the integer abcd has the property a>b>c>d.

abcd - dcba = a 4 digit integer

Repeat over and over and see what happens.

I would also very much like to see a proof for that.

## Answers and Replies

Hurkyl
Staff Emeritus
Science Advisor
Gold Member
if you multiply any number of integers whose digits are only 1, the product will always be palandromic.
That's only true if the numbers are sufficiently small. The proof of when it works is "obvious" from the multiplication algorithm:

Code:
  1111
x  111
------
1111
1111
1111
------
123321

but when the numbers are large enough, you will overflow a digit and will no longer be palindromic. For example, open window's calculator, enter in a number with eleven 1's, and square it.

That's only true if the numbers are sufficiently small. The proof of when it works is "obvious" from the multiplication algorithm:

Code:
  1111
x  111
------
1111
1111
1111
------
123321

but when the numbers are large enough, you will overflow a digit and will no longer be palindromic. For example, open window's calculator, enter in a number with eleven 1's, and square it.

Indeed, it is only palindromic with operations within 9 digits.

if you multiply any number of integers whose digits are only 1, the product will always be palandromic. Grab a calculator and see for your self.

I dont get it. I grabbed a calculator and entered in a number of integers whose digits are only 1.

9*8*7*6*5 = 15120. Thats not palindromic... What am I doing wrong?

cristo
Staff Emeritus
Science Advisor
I dont get it. I grabbed a calculator and entered in a number of integers whose digits are only 1.

9*8*7*6*5 = 15120. Thats not palindromic... What am I doing wrong?

9,8,7,6,5, are integers with only one digit, not numbers whose digits are only 1: i.e. 111 is a number which satisfies the criterion.

ahh, I knew I wasnt comprehending it right. Thank you.

uart
Science Advisor
heres another one, you;ve probably heard of it before.
take a 4 digit integer. All are > 0, and a maximum of 3 of the digits can be equal.

rearrange the digits in such a fashion where the integer abcd has the property a>b>c>d.

abcd - dcba = a 4 digit integer

Repeat over and over and see what happens.

Can someone please give and example of what this one is supposed to do. I'm not seeing anything particularly interesting going on here.

Edit: Ok the digit sum of the result is always 18, is that it?

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If we take a number like: 1244 = 1000 +2*100+4*10+4. Then if we take this Modulo 9, we get 1244 ==1+2+4+4 Mod 9.

So that rearranging these digits doesn't change the value Modluo 9, so that the difference between this number above, and one with the same, but rearranged digits is always divisible by 9.

I don't know if this is what parkmingki had in mind.

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