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Edit $3.$ $-$ I have just found out that the linked competition (see the "Edit $1$.") is still ongoing. Please close and hide this question and its counterpart (see the "Remark."), until the competition is over.

Edit $4.$ $-$ Proposing to reopen the question (the related competition should be over by now).

Edit $3.$ $-$ I have just found out that the linked competition (see the "Edit $1$.") is still ongoing. Please close and hide this question and its counterpart (see the "Remark."), until the competition is over.

Edit $2$. $-$ I've added strategies for solving some simple squares (tiles).

Edit $3.$ $-$ I have just found out that the linked competition (see the "Edit $1$.") is still ongoing. Please close and hide this question and its counterpart (see the "Remark."), until the competition is over.

Edit $2$. $-$ I've added strategies for solving some simple squares (tiles).

Edit $1$. $-$ This problem is equivalent to the problem $20.\space b)$ from Завдання ХХIIІ ТЮМ (2020 р.) which talks about the divisors of $10^{2n}$ where legal moves are $\{\div 10, \times 2,\times 5\}$ to move from the previous divisor to the next one, and you can't revisit a divisor. But, it appears they did not publish the solutions. $-$ Thank you Witold for the reference.

Edit $1$. $-$ This problem is equivalent to the problem $20.\space b)$ from Завдання ХХIIІ ТЮМ (2020 р.) which talks about the divisors of $10^{2k}$ where legal moves are $\{\div 10, \times 2,\times 5\}$ to move from the previous divisor to the next one, and you can't revisit a divisor. But, it appears they did not publish the solutions. $-$ Thank you Witold for the reference.