GOT IT Now

397 /www/nrich/html/content/02/02/15plus2/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> The game <a href="http://nrich.maths.org/public/viewer.php?obj_id=1272&part=index"> Got It</a> is a version of a well known old favourite called Nim. It is an adding game for two. You play against the computer or against a friend. Start with a target of $23$. Set the range of available numbers from $1$ to $5$. Players take turns to add a whole number from $1$ to $5$ to the running total. The player who hits the target of $23$ wins the game. Play the game several times. Can you always win? Can you find a winning strategy? Does your strategy depend on whether or not you go first? Change the game, choose a new GOT IT! target. Test out the strategy you found earlier. Does it need adapting? Can you work out a winning strategy for any target? Is it best to start the game? Always? Change the game again, returning to a target of $23$ but using a different range of numbers this time. Test out the strategies you found earlier. Do they need adapting? Can you work out a winning strategy for any range of numbers? Is it best to start the game? Always? Can you work out a winning strategy for any target and any range of numbers? Extensions: Can you play without writing anything down? Target $24$ using either a $1$, $3$ or $5$. What is the strategy now? Consider playing the game where a player CANNOT add the same as that used previously by the opponent. Play NIM. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Here are some splendid solutions, all from Moorfield Junior School. Well done Moorfield! Each month we get lots of superb work sent in by Moorfield; we hope they'll try some more challenges having made this impressive debut. The first solution is from Adam and Anthony. "We changed the rules by only being allowed to take 1, 2, 3 or 4 and we started with 25 and counted down to 0. We said the key numbers were in the 5 times tables. Your opponent has to go first then whatever your opponent takes you take the amount that adds up to 5. For example: if Mr C took 3 then I would take 2 so it equals one of the key numbers which are 20, 15, 10 and 5." The second is much the same from Adam F and Niall. "In the game we could only take 1, 2 ,3 or 4 cubes and we didn't go first. There were 25 cubes and every time our opponent took a number of cubes we took the number that made it up to 5. The key numbers we had to leave our opponent were: 20, 15, 10, 5." Now a solution from Yuji, Matt and Jimbo all from Moorfield Junior School. "We made our target number 98 and the maximum number we could pick was 7. The first pick was 2, then the computer picked 7 and we picked 1 which made our total 10. Then what ever the computer picked we made it to 8. We did that for 11 times because 8 times 11 equals 88, add to the 10 we made early and that makes 98, the total." The last one is from Steven and Matthew. "We made our target number 98. The highest number we could pick was 7 and the lowest number was 1. The key number was 90 because if we made the total to 90 then the lowest number he could pick was 1 so we would pick 7 to make it 98. We tried different numbers depending on what the computer chose. The numbers when the computer had to choose were: 10, 18, 26, 34, 42, 50, 58, 66, 74, 82 and 90. We tried going first and then going second but we found that going first was the better choice than going second. We made the first total to 10 and then we made it to 8 so the total would be 98." <div class="framework">Editor's note: This is not a fair game because one player can always win whatever the other does. If the target is 43 and the increments are from 1 to 7 then the key numbers are 3, 11, 19, 27, 35 and 43. A player can win by starting with 3, so the first player can always win. For a target of 25 with increments 1, 2, 3 and 4, the first player can't hit a key number, but whatever he chooses, the second player can, so the second player can always win. With a target of 98 and increments from 1 to 7 the first player can start with 2 which is one of the key numbers, so the first player can always win. Even if you know all this it is still a challenging mental arithmetic game if one player chooses the target and range and allows the other player to start, particularly if you play 'against the clock'. In contests of this sort you are not allowed to use the same target and range in a subsequent round.</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Students might be: Thinking strategically in a puzzling context Devising winning ways Following through on insights gained Exploring the notions of complements. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Work backwards. Simplify the game by choosing a smaller target and working out a winning strategy. Investigate complements of $6$. Play the game against the computer and think about the computer's strategy. </mdoxml> 2 5 0 1 1 0 0 GOT IT Now For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target? Numbers and the Number System Factors and multiples Calculations and Numerical Methods Addition & subtraction Using, Applying and Reasoning about Mathematics Generalising