Date: 05/12/97 at 15:22:58 From: Anonymous Subject: A big number? What is the biggest number that can be made using letters only once to make the number? For example, EIGHTY-FOUR. Can anyone find a bigger number? What is the smallest number that can be made? Grant Whitaker
Date: 06/25/97 at 15:19:37 From: Doctor Sydney Subject: Re: A big number? Dear Grant, This is a hard question! We are not sure what the biggest and smallest numbers are that are of the type you describe. That would be a hard problem to solve. However, we do have some ideas for you that might help you think about the solution to the problem. First, think about which numbers from 0 to 9 cannot be used at all because the word representing them contains duplicate letters. For instance, we know that none of our numbers will have a 3 in it when the 3 is positioned such that the 3 has to be said when the number is read. This is because the word three has two e's in it. That means that any number with a 3 in the ones place will not work since when we say all numbers with a three in the ones place we end by saying the word "three" (e.g. 33 = thirty-three). The one exception is numbers that end in 13 - in these cases we say "thirteen," but that too violates the rules for valid numbers because we can't have two e's in our word. So, any number that meets the qualifications specified above will not have a 3 in the ones place. Can a valid number have a 3 in the tens place? Hundreds place? Well, when it is in the tens place, we say thirty-something, right? So, it is possible that we could have a 3 in the tens place -- "thirty-one" works, for example. I'll let you think on your own about whether or not 3 could be in the hundreds place, thousands place, and so on. Be careful! This can be tricky. Continue with bigger numbers. Are there other single-digit numbers that have problems similar to 3's problems? Now, continue by systematically going through words that are used when we say numbers and eliminate ones that are bad on their own, like three, million, trillion, and so on. Then work with the words that are still okay, and try to construct the biggest and smallest numbers with what you have remaining. This could turn into a pretty big job, but you might enjoy it. Just so you know, we did find a number bigger than eighty-four that works: FIVE THOUSAND A small number we found is: MINUS TWO. However, we do not know whether or not these two numbers are the biggest and smallest possible. That is something you should work on! Good luck with this, and thanks for writing! It was sure fun to think about. If have any questions about what we said above or if you have questions about something else, please do write back to us! -Doctors Barney and Sydney, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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