238. Several black and white chips are arranged in a circle. Two players take turns performing the following operation: the first player removes all black chips that have a white neighbor (at least on one side), and the second player then removes all white chips that have a black neighbor. They continue doing this until only chips of one color remain.

a) Suppose there were 40 chips initially. Is it possible that after each player makes two moves, only one chip remains on the circle?

b)* Initially, there were 1000 chips on the circle. What is the minimum number of moves after which only one chip can remain on the circle?
238. Answers: a) yes, b) 8 moves (each player - per move).

Fig. 102 shows an example of arranging 41 chips; next to each chip is its rank-number, indicating how many moves before the end it will be removed. The construction of such an arrangement is convenient "from the end"; to the remaining last black chip of rank 0, add two white chips of rank 1, next to each of them add two black chips of rank 2 (on both sides), next to each black chip add two white chips of rank 3, and so on. It is clear that this method gives the maximum possible number of chips of the corresponding rank at each step. Thus, for each $t$, an arrangement is obtained with the maximum possible number $a_{t}=b_{t}+w_{t}$ of chips - $b_{t}$ black and $w_{t}$ white, which can be transformed into one black chip in $t$ moves ($b_{0}=1$, $w_{0}=0$); the rule for sequential calculation of ($b_{t}, w_{t}$) is very simple: when transitioning from $\boldsymbol{t}$ to $\boldsymbol{t}+1$, the larger of the numbers $\boldsymbol{b}_{t}$, $w_{t}$ does not change, and to the smaller, the doubled larger is added.

| $\boldsymbol{t}$ | 0 | 1 | 2 | 3 | 4 | 5 |  | 7 |  |
| :---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: |
| $b_{t}$ | 1 | 1 | 5 | 5 | 29 | 29 | 169 | 169 | 985 |
| $w_{t}$ | 0 | 2 | 2 | 12 | 12 | 70 | 70 | 408 | 408 |
| $a_{t}$ | 1 | 3 | 7 | 17 | 41 | 99 | 239 | 577 | 1393 |

To solve problem a), it is sufficient to remove one chip of rank 4 from the 41 points in Fig. 102, which does not affect the situation after the first move (i.e., standing next to another chip "4"), for example, to maintain symmetry, the one marked with a star.

The transition from 1000 chips to one cannot occur in fewer than 8 moves, as can be seen from the table, $a_{7}=577<1000$. An example with exactly 1000 chips can be obtained from the maximum arrangement for 8 moves with $a_{8}=1393$ chips, by removing 393 chips of rank 8, which do not affect the situation after the first move - this can be done since $2 \cdot 408=816$ chips of rank 8, placed among 577 chips of rank no more than 7, form no fewer than $816-577=239$ pairs of adjacent chips.