Puzzles and logic › Processes, invariants and strategy

Board operations and invariants

Solve the first problem 18 problems

Sum after repeated difference insertion ●●○○○ 20 ptsWrite the 4 numbers $1,9,8,8$ in a row and perform the following operation: for each pair of adjacent numbers, subtract the left number from the right number, and write the result between the two numbers. This completes one operation. Then perform the same operation on the 7 numbers now in a row, and continue this process for a total of 100 operations. Find the sum of the final row of numbers.
Removing an element to raise probability ●●○○○ 20 ptsTwo different numbers are randomly selected from the set $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$ . The probability that their sum is 12 would be greater if the number $n$ had first been removed from $S$ . What is the value of $n$ ?
Erasing triples on a board ●●○○○ 20 ptsOn a board the numbers $1, 2, 3, \ldots, 27$ are written. One step consists of erasing three numbers $a, b, c$ from the board and writing in their place the number $a+b+c+n$ , where $n$ is a fixed positive integer. Determine the positive integer $n$ such that, after 13 steps, the number $n^2$ is written on the board.
Final number from repeated operation ●●●○○ 50 ptsThe 97 numbers $\frac{49}{1}, \frac{49}{2}, \frac{49}{3}, \ldots, \frac{49}{97}$ are written on a blackboard. Repeatedly pick two numbers $a, b$ on the board and replace them by $2ab - a - b + 1$ until only one number remains. Find the final number.
Product of the two final numbers ●●●○○ 50 ptsThere are a total of 10040 numbers written on a blackboard: 2006 ones, 2007 twos, 2008 threes, 2009 fours and 2010 fives. Each operation erases four different numbers, one of each of four types, and writes one number of the fifth type (for example, erase one 1, one 2, one 3 and one 4, then write one 5; or erase one 2, one 3, one 4 and one 5, then write one 1). After a finite number of operations exactly two numbers remain on the blackboard. Find the product of these two numbers.
Minimum operations to clear 2021 piles ●●●○○ 50 ptsThere are 2021 piles of stones, with the number of stones in each pile being $1, 2, \ldots, 2021$ respectively. In one operation you select any number of piles and remove the same number of stones from each selected pile. What is the minimum number of operations required to remove all the stones?
Reachable values on a blackboard ●●●●○ 100 ptsSvitlana writes the number 147 on a blackboard. Then, at any point, if the number on the blackboard is $n$ , she can perform one of the following three operations: if $n$ is even, she can replace $n$ with $\frac{n}{2}$ ; if $n$ is odd, she can replace $n$ with $\frac{n+255}{2}$ ; and if $n \geq 64$ , she can replace $n$ with $n-64$ . Compute the number of possible values that Svitlana can obtain by doing zero or more operations.
Guaranteed score erasing blackboard numbers ●●●●● 200 ptsThere are 68 pairs of non-zero integers on a blackboard. Some pairs may be identical. For each positive integer $k$ , at most one of the pairs $(k,k)$ and $(-k,-k)$ appears on the blackboard. A student erases some of the 136 numbers so that the sum of any two erased numbers is not $0$ . For each of the 68 pairs, if at least one number from that pair is erased, the student scores one point. Find the largest score that the student can guarantee, whatever the 68 pairs on the blackboard are.
Maximum points removing stones from piles ●●●●● 200 ptsThere are 100 piles, each containing 400 stones. On each move, Petya selects two piles, removes one stone from each, and earns as many points as the absolute difference between the numbers of stones in these two piles (before the removal). Petya must remove all the stones. What is the maximum total number of points he can earn?

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