Interesting little Math problem
February 8th, 2008 | by Sol |Here’s an interesting little problem. I made this one up but I bet it’s been thought of by students of number theory and diophantine equations in particular.
Let’s say you have two kinds of postage stamps, one with a value of 3 cents, the other with a value of 5 cents. List the postage amounts that you CAN’T make. You can’t make 1, 2, 4, or 7 cents.
Let’s look at another example. You have a 5 cent stamp and an 8 cent stamp. What values can’t you make? You can’t make 1, 2, 3, 4, 6, 7, 9, 11, 12, 14, 17, 19, 22, 23, or 27 cents.
For 3 and 5 cent stamps, 7 cents is the largest amount you can’t make.
For 5 and 8 cent stamps, 27 cents is the largest amount you can’t make.
Can you come up with a simple formula for the largest amount you can’t make given two kinds of stamps? What assumption must you make in order for there to be a largest amount you can’t make?
Can you explain why your simple formula works?
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6 Responses to “Interesting little Math problem”
By Jonathan on Feb 9, 2008 | Reply
I think this is great fun. You know the fast food version of this problem, right?
I might offer kids help with this, depending on who they were. Let them stumble a bit, then help organize their investigations: 2 and 11 (easy), then try 3 and 11, then try 4 and 11, then make a conjecture for 5 and 11…double back, if necy, to 5 and 6, 5 and 7…
By MathFr3d on Feb 9, 2008 | Reply
Let x,y be the values of the two stamps
where x== 0}.
By looking sharp you can see that every
x-th number is in the set and for m=1 also
every x+y-th number is in the set. Continuing
so, for m=x-1 every number > (x-1)*y -x is
in the set. So the largest one not in the set
is (x-1)*y -x.
By MathFr3d on Feb 9, 2008 | Reply
Looks like my comment got shreddred because of
bracket signs .. here is my full post:
Let x,y be the values of the two stamps
where x is smaller than y . The largest amount z that
cannot be made using the two kinds of
stamps is:
z = (x-1)*y - x
The formular works because it picks the
largest number that is not in the set
of possible amounts {n*x + m*y | n,m >= 0}.
By looking sharp you can see that every
x-th number is in the set and for m=1 also
every x+y-th number is in the set. Continuing
so, for m=x-1 every number > (x-1)*y -x is
in the set. So the largest one not in the set
is (x-1)*y -x.
By Sol on Feb 10, 2008 | Reply
Jonathan,
No, I had never seen your McNugget version. That’s great! I like your very creative approach to reaching kids. And, I like your approach to helping the kids out a bit and structuring their exploration. That’s something I could use to think about more. I’m learning that I’ve loved Math all of my life and just now I’m starting to think about how to communicate it with others. Thanks for the nudge in that direction.
By Sol on Feb 10, 2008 | Reply
MathFr3d,
I don’t follow your argument but I did get the same answer you did and I have a visual way to show (not rigorously prove) the answer. I’ll make some pretty pictures and post them to the blog.
By krisha on Feb 13, 2008 | Reply
now i can solve my math papers quickly than before. this is really a brilliant version