Competition problem — Combinatorics

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Combinatorics Difficulty 7.9 National olympiad, round 2 Find the answer

Given three letters $X, Y, Z$ , we can construct letter sequences arbitrarily, such as $XZ, ZZYXYY, XXYZX$ , etc. For any given sequence, we can perform following operations:

$T_1$ : If the right-most letter is $Y$ , then we can add $YZ$ after it, for example, $T_1(XYZXXY) =
(XYZXXYYZ).$

$T_2$ : If The sequence contains $YYY$ , we can replace them by $Z$ , for example, $T_2(XXYYZYYYX) =
(XXYYZZX).$

$T_3$ : We can replace $Xp$ ( $p$ is any sub-sequence) by $XpX$ , for example, $T_3(XXYZ) = (XXYZX).$

$T_4$ : In a sequence containing one or more $Z$ , we can replace the first $Z$ by $XY$ , for example,
$T_4(XXYYZZX) = (XXYYXYZX).$

$T_5$ : We can replace any of $XX, YY, ZZ$ by $X$ , for example, $T_5(ZZYXYY) = (XYXX)$ or $(XYXYY)$ or $(ZZYXX).$

Using above operations, can we get $XYZZ$ from $XYZ \ ?$

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

To determine whether we can transform the sequence "XYZ" into "XYZZ" using the operations $T_1, T_2, T_3, T_4,$ and $T_5$ , we systematically examine how these operations affect the sequence:

### Initial Sequence
The starting sequence is:
$\text{XYZ}$

### Available Operations and Their Effects

1. **Operation $T_1$ **: This operation applies only if the sequence ends with "Y":
- Transform $\text{Y} \to \text{YYZ}$ .

Applying $T_1$ to $\text{XYZ}$ , we have:
$\text{XYZ} \to \text{XYZYZ}$

2. **Operation $T_2$ **: This can be applied if the sequence contains "YYY":
- Replace $\text{YYY} \to \text{Z}$ .

However, $\text{XYZ}$ and sequences obtained from it, like $\text{XYZYZ}$ , do not contain "YYY".

3. **Operation $T_3$ **: This can be applied to any sequence:
- Transform $Xp \to XpX$ (append "X" to the end).

Applying $T_3$ to $\text{XYZ}$ , we have:
$\text{XYZ} \to \text{XYZX}$

Applying $T_3$ to $\text{XYZYZ}$ , we have:
$\text{XYZYZ} \to \text{XYZYZX}$

4. **Operation $T_4$ **: Replace the first "Z" with "XY", if any "Z" is present:
- Change $\ldots Z \ldots \to \ldots XY \ldots$ .

Applying $T_4$ to $\text{XYZ}$ , we have:
$\text{XYZ} \to \text{XYXY}$

5. **Operation $T_5$ **: Replace any "XX", "YY", or "ZZ" with "X":
- If these sequences appear, they can be replaced.

However, in sequences like $\text{XYZ}$ , "ZZ" isn't present.

### Analysis

Throughout the available transformations from "XYZ", the creation of a double "ZZ" instead of individual "Z" appears problematic:

- ZZZ Creation: To use $T_5$ to replace with "X", you need "ZZ" specifically. Given our operations, generating a sequence like "ZZ" from transformations and reductions seems out of reach without a replicator, as creating additional "Z" via operators like $T_4$ do not introduce double "Z":

- $T_1$ and $T_3$ extend sequences but do not produce consecutive "ZZ".
- $T_4$ reduces "Z" by $XY$ .

- No operation explicitly compounds consecutive "Z".

Thus, using these operations, the composition of "XYZZ" from "XYZ" is unachievable, concluding:
$\boxed{\text{no}}$

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.