The inhabitants of BZ-land have been cutoff from the rest of the world for generations and their culture has drifted. For example, although they write out numerals in the Arabic manner, when counting verbally to nine, they use “biz” for 5, “buz” for 7 and “baz” for 9. Even more confusing, for numbers above 9, their system requires “biz” to be said once for every time the digit 5 appears in the number, and also an additional “biz” for each time 5 is a divisor of that number. Similar rules exist for numbers that contain and/or are divisible by 7 (“buz”) & 9 (“baz”). Numbers that do not result in any biz-es, buz-es or baz-es merely retain their original name. Hence 10 is spoken “biz”, 27, “buz baz”, and 35 is “biz biz buz”. However, 31 is “thirty-one”. Also, due to their simplicity and isolation, the BZ-landers have no need for numbers greater than 5000 ( “biz biz biz biz biz” ). To them, after that there is just “many”.

One consequence of this system is that verbal counting numbers are not always unique; hence math operations that can be done on paper are hard to interpret out loud. Can you answer the following? Analytical solutions preferred, if they exist. (Full disclosure: the author created the questions and found all answers by creating a list, but believes that some or all of the questions can be answered with analysis, logic and insight).

1) Once you reach the sixth “biz biz biz”, what is the largest number of consecutive numbers that are named by their conventional (English) names before you reach the seventh “biz biz biz”?

2) What is the smallest value that is correctly represented by “biz buz baz”?

3) Is there such a number as “biz biz buz buz baz baz”?

4) Is there such a number as “biz biz biz buz buz baz”?

5) Can you find a sequence of three numbers, n, n+1, n+2 where n=biz, n+1=buz & n+2=baz?

We can see that numbers less than 500 must end in 25, 50 or 55 in order to be biz biz biz. However they can't be multiples of 7, 9, or 125 to avoid any extra biz, baz, or buz. So the sequence would contain 25, 50, 55, 150, 155, 255, 325.

Thus we are looking for a  group of conventional numbers from 255 to 325. It is easy to see that numbers around 315 are not divisible by 5, 7 or 9. The numbers 311-314 are a group of 4 which fit.

315-63=252 is another multiple of 7 and 9, and thus the numbers nearby aren't divisible by 7 or 9. Then, one can see that the midpoint of them is 283.5 and the numbers around them (281-284) are part of of a group of numbers. From looking at the other groups, we can tell there are no more such groups.

Posted by Gamer on 2007-01-18 22:24:24