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Geometry Puzzles

Date: 12/18/95 at 22:13:59
From: Anonymous
Subject: Geometry puzzles

I am puzzled.  I would very much appreciate if you could solve these 
two puzzles I cannot get. In these puzzles you are to determine a 3-digit 
number (no digit repeated) by making "educated guesses."  After each 
guess, you will be given a clue about your guess.  

The clues:  
   bagels: no digit is correct
   pico: one digit is correct but in the wrong position
   fermi: one digit is correct and in the correct position.  

In each of the problems a number of guesses have been made with the clue 
for each guess shown to the right.  Form the given set of clues, and
determine the 3-digit number.  If there is more than one solution, find 
them all.

1. 123  pico, 456 pico, 789 pico, 941 bagels, 375 pico, 638 pico

2. 198 pico fermi, 765 bagels, 432 pico, 129 pico fermi  

Thanks :)

Date: 3/2/96 at 11:18:58
From: Doctor Jodi
Subject: Re: Geometry puzzles

Hi! This is a wonderful puzzle you sent us.  

First of all, if order doesn't matter, and you're choosing 3 distinct 
numbers from 1 to 9, there are 9X8X7 = 504 possible answers, since for 
the first number you can pick any number from 1-9 (9 choices), the 
second time you can choose any number from 1-9 EXCEPT the one you 
picked (8 Choices), and the third time you can choose any number from 
1-9 except the TWO you just picked (7 choices).  Fortunately, the hints 
pare those answers down pretty quickly!

Next, you need to know that for any group of 3 numbers, there are 6 
different ways to write that group.

Say we have the group 1, 2, 3. We can write that as


Now, let's look at the hints.  For each bagel, we know we can't use any 
of those numbers.  For each pico, we MUST use one of those numbers -
but NOT in the place given.  And for each fermi, we MUST use one of 
those numbers in EXACTLY the place given.

So, for 941, a bagel, we can't use any of those numbers.  That leaves us 
with 2, 3, 5, 6, 7, 8 as choices.... But that's still a lot.  Let's see 
what else we can find out about the solution.

In each of the picos, (123, 456, 789, 375, 638), we know that NONE of 
the numbers are in the right place, but that we MUST use exactly one 
number from each group.  Since 3 appears, first, second, and third, we 
can also eliminate it; since it CAN'T be in any of those places, it can't 
be in our final number at all.

Since we just eliminated 1, 3, 4, and 9, and since we must use exactly 
one number from each of the picos, we know that we must use

5 OR 6
7 OR 8
7 OR 5
6 OR 8

We can rewrite this 

if 5, not 6, not 7
if 6, not 5, not 8
if 7, not 5, not 8
if 8, not 6, not 7

Since we must use exactly 3 numbers from 2, 5, 6, 7, 8, and since we 
must use 2 to satisfy the pico 123, we can now see that if we use 5, we 
must use 8 (the only choice left, after using 2 and eliminating 6 and 
7), giving us the group 2, 5, 8

if we use 6, we must use 7 (similarly, the only choice left, after 
using 2 and eliminating 5 and 8), giving us the group 2, 6, 7

We still need to check to see if all of the 12 answers we now have (as 
we said before, 6 from each group of three numbers) will work.  

We still haven't used one fact: none of the numbers in our pico's is 
in the right place.

Remember, those were 123, 456, 789, 375, and 638.

The number 2--can't be middle, so it must be first or last
5--can't be middle or last, so it must be first
6--can't be last or first, so it must be middle
7--can't be first or middle, so it must be last
8--can't be middle or last, so it must be first

Now let's combine this with our groups of numbers--2, 5, 8, and 2, 6, 7 -
to figure out which orders of these numbers will work.

For 2, 5, 8,  8 must be first, 5 must be first, and 2 must be first or 

We can't make 5 and 8 BOTH first at the same time, so NONE of these 
solutions will work.

What about 2, 6, and 7?  2 must be first or last, 6 must be middle, and 
7 must be last.

This gives us the number 267, which is the solution to the first puzzle.

Think you can figure out the second one on your own?

Good luck, and write us back if you need more help.

--Doctor Jodi, The Math Forum

Associated Topics:
High School Logic
High School Puzzles

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