The game of nim comes in a surprisingly wide number of varieties. We analysed the basic version years ago, and celebrated the 200th episode of Problematics last month with a very advanced version involving multiple stacks of objects. Now here is yet another version that requires number awareness of an entirely different sort.
The strategy is based on Fibonacci numbers, that famous series derived from rabbits multiplying from 1 pair to 1, 2, 3, 5, 8, 13, 21… with each term (representing pairs of rabbits) being the sum of the previous two. I shall not dwell extensively on the series here, because many readers will already know it and the rest can check it up on Google or with AI. All I shall say is that the series plays a pivotal part in the strategy for winning the game described below. The game was invented by Robert Gaskell and described by Martin Garden, from whose works I learnt both the rules and the strategy.
#Puzzle 202.1
Two players begin with a specified number of objects, say matchsticks. Each player removes, by turn, a certain number of matches in accordance with the rules. The player who takes the last match or matches wins. The rules are:
(1) The first player can take any number of matches, but not all.
{{/usCountry}}(1) The first player can take any number of matches, but not all.
{{/usCountry}}(2) On subsequent moves, a player can take any number of matches that is not more than twice the number taken by the opponent on his or her previous move.
(3) At any stage after the first move, if the remaining number of matches is twice or less than twice the matches taken by your opponent on his or her last move, you can take all the matches and win the game.
The initial number of matches decides who has the advantage. In most cases, it is the first player who will win by employing the right strategy. Let us take a game that starts with 20 matches, which happens to be a number that favours the first player.
It’s your move, and there are 20 matches on the table. How many matches should you take now to ensure that you win the game in the end?
#Puzzle 202.2
Before the age of spell check, grammar check and AI, newspaper work relied heavily on manual proofreading. Some newspapers, in fact, still have copies read by expert error-spotters before going to print.
Some error-spotters are more expert than others. Real life does not always unfold in percentage terms, but Problematics fiction can do so. We have two proofreaders, neither of them experts, and each has a fixed rate of expertise. That is to say, proofreader #1 will always spot a certain percentage of errors and miss the rest. Proofreader #2 too will always spot a certain percentage of errors, which is fixed as far as she is concerned, but her rate is not the same as that of proofreader #1.
Late one evening, the shift head takes two printouts of the same copy and hands one to each proofreader. Their results are different: proofreader #1 spots 30 errors and proofreader #2 spots only 24. Worried, the shift head asks someone on his staff to compare the two results. It turns out that 20 of the errors have been spotted by both proofreaders.
Suspecting that more errors may lie undetected, the shift head takes it upon himself to check the entire copy from scratch. It finally goes to print all correct.
How many errors went undetected by both proofreaders?
MAILBOX: LAST WEEK’S SOLVERS
#Puzzle 201.1
The answer is: 98 stationery sets were sold, with the total bill being ₹4837.28
With a price of ₹49.36 per set and the bill being ₹XXX7.28, it is clear that the digit in the units place in the number of sets has to be 3 or 8. If the last digit in the number of sets is 3, the digit in the tens place can only be 2 or 7 to get 28 as the last two digits of the product after the number of stationery sets is multiplied by 49.36. Neither of these options results in 7 in the hundreds place of the product.
If 8 is the units digit in the number of sets, the tens place digit can only be 4 or 9 to get 28 as the last two digits of the product. Only 9 results in the next digit of the product being 7. So the answer for the number of sets is 98. The total bill is 98 x ₹49.36 = ₹4837.28.
— Kanwarjit Singh, Chief Commissioner of Income-tax, retired
#Puzzle 201.2
Hello Kabir,
The answers to the anagrams for football players are:
GOOD DAM IN AREA = DIEGO MARADONA
DULL MERGER = GERD MULLER
FRANK SEES CUP = FERENC PUSKAS
GO GET BEERS = GEORGE BEST
SHY ELVINA = LEV YASHIN
The puzzles were relatively easy this time.
— Dr Sunita Gupta, New Dehi
Solved both puzzles: Kanwarjit Singh (Chief Commissioner of Income-tax, retired), Dr Sunita Gupta (Delhi), Sabornee Jana (Mumbai), Shri Ram Aggarwal (Delhi), Anil Khanna (Ghaziabad), Professor Anshul Kumar (Delhi), Shishir Gupta (Indore), Ajay Ashok (Delhi), Vinod Mahajan (Delhi)
Solved #Puzzle 201.1: Yadvendra Somra (Sonipat)
Solved #Puzzle 201.2: YK Munjal (Delhi)