For the following, only letters unique to the spelling of the American English constructed name of the cardinal number shall be counted in determining the length of the cardinal number. Spaces, hyphens and commas are also excluded in determining length. For example, SEVEN has a length of 3 as the E occurs more than once in it spelling. Also note, in American English, the occurance of the connecting word 'and' is ommitted in the construction of a cardinal number's name.

1. What is the largest cardinal number of greatest length?

2. What is the smallest cardinal number of shortest length?

Started with question 2:

ONE = Length 3 (O, N, E)

NINE = Length 2 (I, E)

EIGHTY EIGHT = Length 1 (Y)

THREE-TRILLION, FOUR-HUNDRED, FOUR = Length 0

Now getting down to 0 was tough.  Using ridiculously large numbers like ONE NONILLION ONE it was easy to find a number with length = 0.  However something smaller should work.

Looking at the values TWENTY, THIRTY, . . . , NINETY there is a Y that could only be eliminated if another TWENTY etc was used just prior to a THOUSAND.  So if the answer is less than 20,000 then it could not have one of those numbers with a Y.  First stop: Using HUNDRED.

Now using HUNDRED requires a FOUR to eliminate the U, THREE or EIGHT (or THIRTEEN or EIGHTEEN) to eliminate the H.  But any of these combinations would then require another number to get rid of the F and/or the I, and this would take us above three digits and thus into the realm of THOUSAND.

Now THOUSAND has an A which is not found anywhere again unless we us QUADRILLION.  Looks like we have to move on to MILLION.

And MILLION has an M which for our purposes is not used in any number used in polite society.

And BILLION has a B which is not used in any other number

So lets go for TRILLION!

TRILLION = 4 (T, R, O, N)

The R requires a THREE, FOUR, THIRTEEN, FOURTEEN, THIRTY, FORTY or HUNDRED

Working with ONE TRILLION = 3 (T, R, E) and

ONE TRILLION THREE = 1 (H)

It should be easy to get rid of the H, but it requires another THREE or THIRTEEN, or THIRTY (EIGHT, EIGHTEEN, or EIGHTY), which adds a Y or forces a HUNDRED.

Since HUNDRED works better that THIRTY . . .

ONE TRILLION THREE HUNDRED = 1 (U)

This requires a FOUR or FOURTEEN to eliminate the U

ONE TRILLION THREE HUNDRED FOUR = 1 (F)

Need another FOUR, FOURTEEN or FORTY, but THOUSAND MILLION and BILLION will not work so rework to

THREE TRILLION, FOUR HUNDRED, FOUR

Posted by Leming on 2008-09-28 20:19:47