Change for a Dollar
Math brain teasers require computations to solve.
Assuming you have enough coins of 1, 5, 10, 25 and 50 cents, how many ways are there to make change for a dollar?
Comments
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Posted by Nike01 on Aug 24, 2001 wrong...192 |
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Posted by Mogmatt16 on May 02, 2002 explane please |
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Posted by dewtell on Jul 25, 2002 I get 292. Let N(a, c) be the number of ways to represent an amount a using coins no larger than value c. Then N(a, 1) = 1 (there is only one way to represent any amount using only pennies), and N(0, c) = 1 (there is only one way to represent an amount of zero, no matter what coins are available). N(5n, 5) = n+1 (you can use anywhere from 0 to n nickels, and then have to make the rest up in pennies). N(10n, 10) = (n+1)^2 (the sum of N(0, 5), N(10,5), N(20, 5), ..., N(10n, 5), which is the sum of the first n+1 odd numbers - it's an old fact that this sum yields the perfect squares). N(10n+5, 5) = (n+1)*(n+2). Now go top down. We want N(100, 50). N(100, 50) = N(0, 25) + N(50, 25) + N(100, 25), representing the use of 2, 1, and 0 fifty-cent pieces, respectively. N(50, 25) = N(0, 10) + N(25, 10) + N(50, 10). N(100, 25) = N(0, 10) + N(25, 10) + N(50, 10) + N(75, 10) + N(100, 10). Using our previous facts about N(a, 10), we find that N(100, 25) = 1 + 12 + 36 + 72 + 121 = 242; N(50, 25) = 1 + 12 + 36 = 49; and finally, N(100, 50) = 1 + 49 + 242 = 292. |
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Posted by dewtell on Jul 25, 2002 Oops - reference to N(10n+5, 5) in the previous comment should be N(10n+5, 10) instead. |
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Posted by othbrady on Mar 07, 2003 good 1 |
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Posted by GTregay on Jan 21, 2005 I didn\'t take the time to follow the logic on this one - created an Excel spreedsheet, and it is indeed 292!  |
Posted by darthforman on Apr 19, 2005 My head hurts. It's the right answer but I don't really like the math ones. It takes too much work.  |
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Posted by tdc_tmc on Aug 28, 2005 woooooooow!!!!! too much for me  |
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Posted by hidentreasure on Nov 29, 2005 Techer told me a while back. good teaser.   |
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Posted by irar1792 on Dec 31, 2005 AAAAAAAAAAAAAAAAAAggggggggghhhhhhhhhhhhh  |
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Posted by blackmarket69 on Mar 08, 2006 coins are shiny!! i like things that r shiny |
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Posted by Methlos on Mar 12, 2006 it would be the same if you used GB pence |
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Posted by spazzyjazzy93 on Apr 05, 2006 i got that but only because i read that fact on a snapple cap |
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Posted by righteouschick2 on Apr 08, 2006 who has time for that stuff?¿? |
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Posted by banzai on Jul 03, 2006 my answer was 'alot"! |
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Posted by hsbanana90 on Aug 20, 2006 that was hard. it would take forever to figure that out, unless you knew how to do the calculator functions. |
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Posted by josty on Aug 27, 2006 easy but long |
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Posted by ari_1 on Sep 04, 2006 wooooah .....umm i got that wrong!!  |
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Posted by teasermaster on Oct 13, 2006 Way to tough for me! |
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Posted by kristybubba on Dec 06, 2006 people call me the smart goddess of math but i aint that smart it hurts my head   |
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Posted by kristybubba on Dec 06, 2006 people call me the smart goddess of math but i aint that smart it hurts my head |
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Posted by QTkate on Jan 15, 2007 um yeah well i agree with evryone else that was 2 hard foe my lil head |
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Posted by bgil7604 on Feb 23, 2007 Wow, you must have WAY too much time on your hands if you know that answer! |
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Posted by Brock on Jul 30, 2007 nice 1  |
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Posted by Brock on Jul 30, 2007 nice 1 |
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Posted by SuperDude512 on Jul 31, 2007 wow i jus realized how bad i am at math, NICE TEASER KID! |
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Posted by Xosyaraiha on Aug 08, 2007 they tried to ask that to my math class... and didn't succeed. glad SOMEONE could get the answer thumbs up |
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Posted by srpwuzhere on Nov 23, 2008 I watched a Spongebob Squarepants episode where Spongebob was talking about change for $1 and $5. hehe. |
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Posted by javaguru on Jan 11, 2009 I did it similar to dewtell, slightly less formally. The real break is between using dimes and smaller and using quarters and larger. So I broke the problem into making 100, 75, 50, 25 or 0 cents from dimes and smaller. To make 100 cents from dimes and smaller start with 10 dimes and replace from 0 to 10 dimes with two 5s. Each 5 can be made from either a nickel or 5 pennies, so there is 1 way to replace 0 dimes, 3 ways to replace 1 dime, 5 ways for 2 dimes... giving the sum of the first 11 odd numbers, which as dewtell points out is 11^2 = 121. So making 100 cents with only dimes or less is 121. 75 cents has a 5 and 7 10s. From 0 to 7 10s can be replaced with two 5s. The extra 5 means that there is one extra way to make each value (because the first 5 can either be a nickel or 5 pennies). This gives 8^2 + 8 = 72 ways to do this. There is one way to make the 25 so there are 72 ways with 75 cents made from dimes and smaller. 50 cents is 6^2 = 36 times the two ways to make the 50 (50, 25 x 2) = 72. 25 cents is 3^2 + 3 = 12 times the two ways to make 75 (3x25, 50+25) = 2 x 12 = 24. There are 3 ways to make a dollar with no dimes and pennies (2x50, 50 + 2x25, 4x25) = 3. 121 + 72 + 72 + 24 + 3 = 292 |
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Posted by Jessi-wa on Oct 20, 2011 urgh.... math makes my head hurt soooo bad |
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