Problematics | Games with matches and jumbled scientists
Here is an ingenious variation of nim, the game in which two players remove matchsticks from the table. Can you identify the winning strategy?
While planning for your puzzles for the week, I came across an excellent variation of the well-known game ‘Nim’. Problematics readers have solved at least one more version earlier, but I shall describe the basic rules nevertheless for those who missed that episode and those who are not familiar with the game.
In the basic version of nim, two players start with a previously agreed number of matchsticks, ludo counters or any other suitable objects. Say there are 25 matches. Each player can remove, by turn, one, two or three matches. The player who takes the last match or matches wins.
If the starting number is not a multiple of 4, the first player can always win if they know the strategy, which is now almost as well-known as the game itself. Simply take as many matches as are necessary to reduce the remainder to a multiple of 4. In the example above, take away one match so that the remainder becomes 24; if you started with 30, take away two matches to bring the remainder to 28. After that, at every move, once the second player has taken away one, two or three matches (the maximum), the first player repeats the strategy. They continue this way until only four matches remain; whatever number the second player takes now, the first player sweeps away the rest.
If you start with a multiple of 4, the first player’s opening move will have no option but to take one, two or three matches, leaving a remainder that is not a multiple of 4. The second player now follows the above strategy and wins.
The variation described below was created by Robert F Gaskell, as acknowledged by the late Martin Gardner.
#Puzzle 67.1
Once again, you start with a certain number of matches. And once again, the player to remove the last match or matches wins. The number you can remove this time, however, is almost unlimited. Either player can remove as many matches as they like, subject to two restrictions.
One, the first player cannot remove all the matches on their opening move, but can remove any other number. On subsequent moves, either player can remove any number of matches, including all of them if the second restriction permits it.
The second restriction is after the opening move, you cannot take more than twice the number of matches your opponent removed on the previous move. For example, if your opponent has just removed four matches, you may not remove more than eight.
Here again, the first player has the advantage if they know the winning strategy. To give an example, let us start with 100 matches. The first player mentally breaks up 100 = 89 + 8 + 3 and picks up the smallest of these constituents, or 3 matches. The second player cannot take more than 6. Suppose they take 5 matches. This leaves 92 matches. The first player now calculates 92 = 89 + 3 and takes way 3.
This leaves 89. Suppose the second player this time takes the maximum they are allowed, which is twice 3 (the first player’s previous move), or 6. This leaves 83, which the first player breaks up as 83 = 55 + 21 + 5 + 2, and takes away 2. And so on, until the first player wins.
The above example gives enough hints about the kind of numbers that the first player is interested in.
What is their winning strategy?
#Puzzle 67.2
I made the following anagrams out of the names of 10 scientists, with a little bit of innovation: each anagram includes two names. The word “AND” is part of each anagram.
Can you unravel them all?
MAILBOX: LAST WEEK’S SOLVERS
#Puzzle 66.1
Hi,
The table shows who bowls what in each round.
— Ajay Ashok, Mumbai
#Puzzle 66.2
Dear Kabir,
The secret lies in the following mathematical principle:
(True) x (False) = (False)
Therefore, one must smartly ask a question that would evoke a “compound” of answers from both robots. From the stated principle, we know that the outcome would always be a false statement and then we can deduce the truth correctly.
So, ask the following question to either of the robots:
“If I were to ask the other robot ‘What is your name?”, what answer would I get?”
Irrespective of whether this robot tells the truth or not; the reply is going to be false. Therefore, if you get the answer as ‘A’, the name of this Robot is ‘B’, and vice versa.
— Group Captain RK Shrivastava (retired), Delhi
This is the answer I had in mind, having based the puzzle on an existing one which some readers recognised. What most readers, and I, failed to observe was that there is a simpler solution. If you remember, the original question required you to find out an unknown direction with a single question. Here, you only need to identify the two names, and only two readers offered an alternative, very effective solution. One is Geetansha Gera, the other is below:
Hi Kabir,
A question with a known answer, such as “How many days in one week?” can identify A or B.
— Shishir Gupta, Indore
Solved both puzzles: Ajay Ashok (Mumbai), Group Captain RK Shrivastava (retired; Delhi), Shishir Gupta (Indore), Geetansha Gera (Faridabad), Dr Sunita Gupta (Delhi), Akshay Bakhai (Mumbai), Yadvendra Somra (Sonipat), Professor Anshul Kumar (Delhi), Amardeep Singh (Meerut)
Solved #Puzzle 66.2: HP Choudhury (Greater Noida)
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