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### Mensa: Numbering for an Alternate World

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Date: 15 May 1995 16:42:58 -0400
From: Mary Wernke
Subject: Mensa Puzzler

In a parallel universe, the numbering system in use is based on the
26-character Roman alphabet. A is the number ), B is the number 1, C is
the number 2, and so on, through Z is the number 25.

You are in the parallel universe and are driving from New York to San
Francisco. A road sign indicates you are BBQ miles from San Francisco.
Question number one: Are you closer to Reno, VN; Salt Lake City, UT or
Kansas City, MO?

Once in San Francisco, you stop for a plate of ribs, fries and a shake.
Because there's a lunar eclipse in progress, the manager is offering a
free meal to anyone in the restaurant who can tell him the distance from
the Earth to the moon at its closest point. You think you remember the
distance to the moon at perigee is 220,753 miles, but obviously you have
to tell him the answer in this strange numbering system. Question number
two: What would the answer be?
```

```
Date: 16 May 1995 23:34:12 -0400
From: Dr. Sydney
Subject: Re: Mensa Puzzler

Hello!

Thanks for writing Dr. Math!

This problem revolves around an understanding how different bases
work. Have you dealt with bases very much?  People (and more often
computers) work with base 2 a lot.  In base 2 you use only 2 digits, 0 and 1.

The different base systems all revolve around one main unifying concept.
To understand it, let's first look at how it works in base 10 (the system we
use!).  When we write a number down in base 10, like the number 2876, we are
really using a shorthand notation -- what we mean is that we have 6 1's (or
10^0's), 7 10's (or 10^1's), 8 100's (or 10^2's) and 2 1000's (or 10^3's).
So we are really saying that 2876 = 6*10^0 + 7*10^1 + 8*10^2 + 2*10^3.

Similarly, given the number 1101 in base 2, we are really saying that

1101 (base 2) = 1*2^0 + 0*2^1 + 1*2^2 + 1*2^3

More generally, given base b,

a number xyz (base b) = z*b^0 + y*b^1 + x*b^2

In your problem we are dealing with base 26.
To figure out what BBQ (base 26) is, write it out as in the above examples...

BBQ(base 26) = Q*26^0 + B*26^1 + B*26^2

(you'll have to fill in the number equivalents for the letters!!).
You'll get a number that will help you to answer that first question.

Now, your second question asks you to convert the number 220,753 (base 10)
to base 26.  This is a bit trickier.  We'll do a process that is kind of the
reverse of what we did above.  Let's look at the problem on a simpler
level...say we are given the number 25 in base 10 and want to know what it
is in base 3.  Well, we must first consider all the powers of 3 (3^0, 3^1,
3^2, 3^3, ...)  What is the largest of these that is still less than 18?
It looks like 3^2 (which is 9) is the largest power of three that is still
less than 25 (3^3 = 27 which is too big!).  So, how many 3^2's are there in
the number 25?  Well, we can fit 2 9's in the number 25, and when we do that
we have a remainder of 7 (this is because 25 - 2*3^2 = 7).  This may seem
familiar...it is like long division, really.  So,anyway, that means we put a
2 in the 3^2 place.  Now, we look at the 3^1 place.  How many 3's are there
in 7?  Well, there are 2, so we put a 2 in the 3^1 place.  We have 1 left
over (7- 3*2 = 1 ), and we move on to the 3^0 place (the ones place!).  We
have 1 left, so we put a one in the ones place.  Thus, we've found that
25 (base 10) = 221 (base 3).  You can check that this is the right answer by
verifying that 221 (base 3) = 25 (base 10) using methods discussed earlier.

Okay, so now that we can convert with smaller bases, we just apply the
same ideas to the problem below with base 26.  To start with, write out the
powers of 26 (26^0, 26^1, 26^2...).  What is the largest of these that is
still less than 220,753?  Now, just do what we did above, and before you
know it you'll have an expression for 220,753 in base 26.

feel free to write back!

--Sydney, "dr. math"
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Associated Topics:
High School Number Theory
High School Puzzles

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