### Brain Teasers

# Change for a Dollar

Math
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?

### Hint

There are more than 200.### Answer

There are 292 ways.Hide Hint Show Hint Hide Answer Show Answer

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## Comments

wrong...192

explane please

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.

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.

Oops - reference to N(10n+5, 5) in the

previous comment should be N(10n+5, 10)

instead.

previous comment should be N(10n+5, 10)

instead.

good 1

I didn\'t take the time to follow the logic on this one - created an Excel spreedsheet, and it is indeed 292!

My head hurts.

It's the right answer but I don't really like the math ones. It takes too much work.

woooooooow!!!!! too much for me

Techer told me a while back. good teaser.

AAAAAAAAAAAAAAAAAAggggggggghhhhhhhhhhhhh

coins are shiny!!

i like things that r shiny

i like things that r shiny

it would be the same if you used GB pence

i got that but only because i read that fact on a snapple cap

who has time for that stuff?Â¿?

my answer was 'alot"!

that was hard. it would take forever to figure that out, unless you knew how to do the calculator functions.

easy but long

wooooah .....umm i got that wrong!!

Way to tough for me!

people call me the smart goddess of math but i aint that smart

it hurts my head

it hurts my head

people call me the smart goddess of math but i aint that smart

it hurts my head

it hurts my head

um yeah well i agree with evryone else that was 2 hard foe my lil head

Wow, you must have WAY too much time on your hands if you know that answer!

nice 1

nice 1

wow i jus realized how bad i am at math, NICE TEASER KID!

they tried to ask that to my math class... and didn't succeed. glad SOMEONE could get the answer

thumbs up

thumbs up

I watched a Spongebob Squarepants episode where Spongebob was talking about change for $1 and $5. hehe.

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

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

urgh.... math makes my head hurt soooo bad

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