An amazing pattern!

Take any number $m$ (let's say 3).
Write all the numbers starting from $1$ in one line.
Eliminate numbers at the index $k$ ( $k \equiv 0 \pmod{m}$ )
Now write for each index, partial sum of the numbers till that index.
Eliminate numbers at the index $k$ ( $k \equiv m-1 \pmod{m}$ ).
Now write for each index, partial sum of the numbers till that index.

$\vdots$

$\quad$ 2m-1. Eliminate numbers at the index $k$ ( $k \equiv 2 \pmod{m}$ ).

$\quad$ 2m. Now write for each index, partial sum of the numbers till that index.

For example, for $m=3$ ,

$\begin{array}{ c c c c c c c c c c c c c} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13\\ 1 & 2 & & 4 & 5 & & 7 & 8 & & 10 & 11 & & 13\\ 1 & 3 & & 7 & 12 & & 19 & 27 & & 37 & 48 & & 61\\ 1 & & & 7 & & & 19 & & & 37 & & & 61\\ 1 & & & 8 & & & 27 & & & 64 & & & 125\\ \end{array}$

$\displaystyle \text{We get a sequence of} \ n^m \text{!}$

Prove that this happens for any number of numbers and for any natural number $m$ .

PS - This is not a challenge. This is seeking help.

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print "hello world"

Math

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Remember to wrap math in $ ... $ or \[ ... \] to ensure proper formatting.

2 \times 3

2 \times 3

2^{34}

2^{34}

a_{i-1}

a_{i-1}

\frac{2}{3}

\frac{2}{3}

\sqrt{2}

\sqrt{2}

\sum_{i=1}^3

\sum_{i=1}^3

\sin \theta

\sin \theta

\boxed{123}

\boxed{123}

Easy Math Editor

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

(1) I have fixed your LaTeX. The correct code here is "\begin{array}".

(2) This result is known as Moessner's Theorem.

Jon Haussmann - 3 years, 10 months ago

Oh I see. Thanks! I will look into the references. BTW, can you give a simplified proof?

Kartik Sharma - 3 years, 10 months ago

@Jon Haussmann

I'm afraid I don't know the proof. I think it's a hard result.

@Jon Haussmann – Yeah. I am not able to find an understandable proof yet. And its generalizations to the way Conway has done in his book, is magical indeed.

@Mark Hennings @Pi Han Goh @Ishan Singh Please also fix the latex!

This is seriously magic. Another math-e-magical thing with which you can fascinate a primary school student and can "irritate" a mathematician.

(Another such thing is the Goldbach's Theorem).

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$