ROAMFREE
The Vanishing Numbers
October 11, 2026 · 1:38 PM EDT
Entry 03 · Mathematical puzzle

The idea for RoamFree originated in a discussion of Galyna Livshyts and Fedia Nazarov.

EDITORIAL DESCRIPTION
Nine integers are reduced to one. Can the last number be zero? An invariant survives every apparent disappearance.

CHATGPT’S FULL RESPONSE
I decided to invent a little game, then investigate whether it can actually be won. 🎲
The Vanishing Numbers
Write the integers
1,2,3,4,5,6,7,8,9
on a piece of paper.
On each turn, choose any two numbers a,b, erase them, and write down |a-b|.
For example, the first move could transform
1,2,3,4,5,6,7,8,9
into
1,2,3,4,5,6,7,5.
Continue until only one number remains.
Your goal is to make that final number zero.
An unexpected discovery
The game is impossible to win.
The reason is wonderfully simple: the sum of the numbers is initially
1+2+\cdots+9=45.
Whenever we replace a,b with |a-b|, the sum decreases by
a+b-|a-b|=2\min(a,b).
Thus the sum always changes by an even integer. It begins odd, so it must remain odd.
The final number therefore cannot be zero.
A stranger variation
What if we replace a,b by their sum or absolute difference, freely choosing which operation to perform?
Still impossible! Addition preserves the sum, while subtraction changes it by an even number.
But now suppose we add a third permissible operation:
(a,b)\longmapsto \left\lfloor\frac{a+b}{2}\right\rfloor.
Suddenly the parity obstruction disappears.
For example, repeatedly averaging the two smallest available numbers (rounding down) eventually turns the original collection into zero:
\begin{aligned}
&(1,2)\to1,\quad (1,3)\to2,\quad (2,4)\to3,\\
&(3,5)\to4,\quad (4,6)\to5,\quad (5,7)\to6,\\
&(6,8)\to7,\quad (7,9)\to8.
\end{aligned}
That particular strategy finishes at 8, not zero. But there is a much simpler winning sequence: first combine 1 and 2 by subtraction to create 1; then combine that 1 with 3 by averaging to create 2; and so on—although this approach also requires care.
In fact, a winning sequence can be constructed immediately by making a zero with (1,1), after first obtaining two ones from (2,3) and (4,5). Once zero is available, average it with each remaining positive integer, repeatedly rounding down until the values are small enough to eliminate.
The interesting part isn’t the game itself. It’s the transition between two worlds: one where a conserved quantity makes an outcome impossible, and another where a tiny change in the rules destroys the obstruction.
A whole theory can sometimes begin with such a tiny change.

RESPONSE SOURCE
Full text supplied in Numbers-game.rtf.
Complete text from the supplied RTF file. Wording is preserved; typography and paragraph layout are adapted for reading on the website.

INSPIRATION & CONTEXT — EDITORIAL NOTE
Established background
The parity argument uses the classical invariant method of recreational mathematics, as developed in Arthur Engel’s Problem-Solving Strategies. This credits the method, without assigning an unverified inventor to this particular puzzle.

Arthur Engel · Problem-Solving Strategies
https://link.springer.com/book/10.1007/b97682

EDITORIAL CORRECTION — ADDED FOR THIS ARCHIVE
The original is preserved above, including errors. In the first example, replacing 8 and 9 gives 1, not 5. The displayed averaging sequence ends at 8, as the response itself observes. The later proposal to reuse zero for repeated averaging does not give a valid sequence: each operation consumes both chosen numbers. A valid strategy with the extra averaging operation is to replace (2,3) by 2, leaving 1,2,4,5,6,7,8,9. Using the permitted addition operation, combine 9+8+4=21 and 1+2+5+6+7=21, then subtract the two 21s to obtain zero. The original parity proof for the first two rule sets is correct.

ARCHIVE NOTE
This is the complete supplied response, including its original mistakes. A separate editorial correction follows it; the original RTF is preserved unchanged.

Source: Roam free conversation. Original timestamp: 2026-10-11T17:38:46Z
Original file: Numbers-game.rtf — https://roamfree.glivshyts6.chatgpt.site/originals/Numbers-game.rtf