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 Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion

 May 31st, 2015, 07:29 PM #1 Senior Member   Joined: Oct 2013 From: Far far away Posts: 429 Thanks: 18 The maximum impossible score Question: In a dart game, only 4 points or 9 points can be scored on each dart. What is the largest score that it is NOT possible to obtain? (Assume that you have an unlimited number of darts) My attempt: I listed the impossible scores, as follows... 1, 2, 3, 5, 6, 7, 8, 10, 11, 14, 15, 19, 23, s After 23, it appears that all possible numbers can be expressed as a combination of 4's and 9's. Is 23 the answer? As you can see, my method doesn't allow me to know that 23 is the maximum. It only suggests 23 as an answer. Is there a better more decisive method to solve this problem? Thanks May 31st, 2015, 09:19 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra GCD(4,9) = 1 which means that there exist $x,y \in \mathbb Z$ such that $4x + 9y = 1$. In fact we have $x = -2$ and $y=1$. Thus, if we can make a number $n$ using 2 or more 4s, we can also make $n+1$. In the light of this, does the following tell you anything? 24 = 6 x 4 25 = 4 x 4 + 9 26 = 2 x 4 + 2 x 9 27 = 0 x 4 + 3 x 9 Thanks from shunya and Country Boy Tags impossible, maximum, score Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post phill1187 Advanced Statistics 4 October 31st, 2014 11:33 AM jskrzy Algebra 2 November 3rd, 2011 10:27 AM saxoxymoron Algebra 0 June 29th, 2011 02:56 AM akira357 Advanced Statistics 1 March 4th, 2010 02:42 PM finitehelp Probability and Statistics 1 August 4th, 2009 07:22 AM

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