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Omni-MATH / 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: T₁: If the righ…

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Given three letters X,Y,ZX, Y, Z, we can construct letter sequences arbitrarily, such as XZ,ZZYXYY,XXYZXXZ, ZZYXYY, XXYZX, etc. For any given sequence, we can perform following operations: T1T_1: If the right-most letter is YY, then we can add YZYZ after it, for example, T1(XYZXXY)=(XYZXXYYZ).T_1(XYZXXY) = (XYZXXYYZ). T2T_2: If The sequence contains YYYYYY, we can replace them by ZZ, for example, T2(XXYYZYYYX)=(XXYYZZX).T_2(XXYYZYYYX) = (XXYYZZX). T3T_3: We can replace XpXp (pp is any sub-sequence) by XpXXpX, for example, T3(XXYZ)=(XXYZX).T_3(XXYZ) = (XXYZX). T4T_4: In a sequence containing one or more ZZ, we can replace the first ZZ by XYXY, for example, T4(XXYYZZX)=(XXYYXYZX).T_4(XXYYZZX) = (XXYYXYZX). T5T_5: We can replace any of XX,YY,ZZXX, YY, ZZ by XX, for example, T5(ZZYXYY)=(XYXX)T_5(ZZYXYY) = (XYXX) or (XYXYY)(XYXYY) or (ZZYXX).(ZZYXX). Using above operations, can we get XYZZXYZZ from $XYZ \ ?$
Plain-text mathematical notation (without MathML)
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:

T₁: If the right-most letter is Y, then we can add YZ after it, for example, T₁(XYZXXY)=(XYZXXYYZ).

T₂: If The sequence contains YYY, we can replace them by Z, for example, T₂(XXYYZYYYX)=(XXYYZZX).

T₃: We can replace Xp (p is any sub-sequence) by XpX, for example, T₃(XXYZ)=(XXYZX).

T₄: In a sequence containing one or more Z, we can replace the first Z by XY, for example,
T₄(XXYYZZX)=(XXYYXYZX).

T₅: We can replace any of XX,YY,ZZ by X, for example, T₅(ZZYXYY)=(XYXX) or (XYXYY) or (ZZYXX).

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

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Source and history

task_id
17b7ec60-f6ce-5490-862b-0c1f98397901
task_key
test--17b7ec60-f6ce-5490-862b-0c1f98397901
task_revision_id
1
config / split
/ test

Official source

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