This is a slide show to explain in detail how to solve a puzzle of a common sort. Use the right...

This is a slide show to explain in detail how to solve a puzzle of a common sort. Use the right arrow key to go to the next step and left arrow keys to go to the last step. To terminate before the end, hit the Escape key. Ready? Hit the right arrow key.

To solve this problem, we should try to find a way to organize the information so that drawing inferences is easy or even mechanical. Let us set up some tables to represent all of the important features of the problem. Hit the right arrow key to continue.

High in the mountains of a wilderness park, four backpackers happened to meet at the same campsite, where they all spent the night. One manwas from Billings. From the statements below, determine the hometown of each man and the color of his backpack.

1. Larry had a fire made before the man from Tucson and the fellowwith the blue pack arrived. Kevin came even later.

2. Both Perry and the man from San Francisco said it was their firsttrip to this spot. The man with the red pack said he had been there twice before.

3. The man from L.A. gave Fred and the person with the green packsome moleskin for their blistered feet. "None for me, thanks," saidKevin, "It's my back that hurts."

4. "I believe I've travelled the farthest to get here," said the manwith the orange pack. "No," Fred replied, "I think Kevin did."

With these we will be able to keep track of the information without having the puzzle set up in front of us. If we find, for instance that Larry came from L.A., then we’ll put an ‘X’ in the square that goes down the ‘Larry’ column and across the ‘L.A.’ row. If we find that Larry is not from L.A., then we’ll put a “~” in that square. Hit the right arrow key to continue.

Larry Perry Fred Kevin blue orange red green

Billings BillingsSan Fran San FranL.A. L.A.Tucson Tucson

blueorangeredgreen

Larry Perry Fred Kevin






Let’s take statement number one and see what we can infer from it. It seems that Larry cannot be from Tucson or have blue pack, for no one can make a fire in a place before they arrive at that place. Furthermore Kevin must also be different from the man from Tucson and not have a blue pack. So let us put the appropriate marks in the squares to note these 4 facts. Hit the right arrow key to continue.


Billings BillingsSan Fran San FranL.A. L.A.Tucson Tucson

blueorangeredgreen







Larry is not from Tucson and Larry doesn’t have a blue pack and Kevin is not from Tucson and Kevin does not have a blue pack. These 4 pieces of information account for the 4 tildes in the boxes. But there is something else we may be able to infer from 1. It doesn’t strictly follow but the problem cannot be done without inferring something from the context. The man from Tucson is not identical to the man with the blue pack. A logician would not talk this way so let us assume that the writer of the puzzle was not a logician. Hit the right arrow key to continue.


Billings BillingsSan Fran San FranL.A. L.A.Tucson ~ ~ Tucson

blue ~ ~orangeredgreen







So we put a ‘~’ in the Tucson/blue square. Now let’s take the second statement and proceed in the same way. Hit the right arrow key to continue.


Billings BillingsSan Fran San FranL.A. L.A.Tucson ~ ~ ~ Tucson





2. Both Perry and the man from San Francisco said it was their firsttrip to this spot. The man with the red pack said he had been there twice before



Using the same sort of logic we can tell that Perry is not from San Fran and the man with the red pack is not Perry or from San Fran. So let’s mark those 3 squares with tildes. Hit the right arrow key to continue.


Billings BillingsSan Fran San FranL.A. L.A.Tucson ~ ~ ~ Tucson








Perry is not from San Fran and the man with the red pack is not Perry or from San Fran. Now let’s consider the 3rd statement. Hit the right arrow key to continue.


Billings BillingsSan Fran ~ ~ San FranL.A. L.A.Tucson ~ ~ ~ Tucson

blue ~ ~orangered ~green







From #3 we see that Fred is not from L.A., Fred does not have a green pack and the Man from L.A. does not have a green pack. Furthermore Kevin is not from L.A. or have a green pack. Let’s fill in the squares with tildes. Hit the right arrow key to continue.


Billings BillingsSan Fran ~ ~ San FranL.A. L.A.Tucson ~ ~ ~ Tucson

blue ~ ~orangered ~green







That’s why we got 5 new tildes (in red). Now let’s consider statement #4 Hit the right arrow key to continue.


Billings BillingsSan Fran ~ ~ San FranL.A. ~ ~ ~ L.A.Tucson ~ ~ ~ Tucson

blue ~ ~orangered ~green ~ ~







What can we infer from 4? That Fred and Kevin do not have an orange pack. Let’s put in those two new tildes. Hit the right arrow key to continue.



blue ~ ~orangered ~green ~ ~







Fred and Kevin do not have an orange pack. So we got 2 new tildes (in red). We now have captured all of the important information in the boxes. We can get rid of the statement of the original puzzle and see what we can reason. Do you see any way to make inferences. Think about it before you hit the right arrow key.



blue ~ ~orange ~ ~red ~green ~ ~







At this point I will introduce a couple of useful principles. First of all, (Principle 1), if you have all but one square in a row or column filled with tildes, then the empty square should get an ‘X’. Hit the right arrow key to continue.



blue ~ ~orange ~ ~red ~green ~ ~


So since Kevin does not have a blue or orange or green backpack, we’ve inferred that Kevin has a red backpack. But if Kevin has the red backpack then the other guys don’t have that color pack. So the second principle is that if you have an ‘X’ in a row or column then tildes go in all the other squares of that row or that column. Imagine how we will do that and then hit the right arrow key to continue.



blue ~ ~orange ~ ~red ~ Xgreen ~ ~


We only got two new tildes because one tilde was already on that row. Note that because of one of those tildes we now have 3 tildes on another column. So we can put in another ‘X’, Find it before you hit the right arrow key to continue.



blue ~ ~orange ~ ~red ~ ~ ~ Xgreen ~ ~


And this new ‘X’ mark lets us fill in one more tilde. Let’s do that. Hit the right arrow key to continue.



blue ~ X ~orange ~ ~red ~ ~ ~ Xgreen ~ ~


Before we can continue we need at least one (or two more principles). There are no more ‘X’s or ‘~” to be made from our first two principles. So consider this: If Fred is identical to the man with a blue pack and the man with the blue pack is not from Tucson (both facts seen above), then what can we infer? Hit the right arrow key to continue.



blue ~ ~ X ~orange ~ ~red ~ ~ ~ Xgreen ~ ~


If Fred is identical to the man with a blue pack and the man with the blue pack is not from Tucson then Fred is not from Tucson (red tilde above). The general principle is if a is identical to b and b is not identical to c then a is not identical to c. We can also see that if a=b and b=c then a does = c. So these two (or one) principles are enough with the others to solve the rest of this puzzle. There are many ways to go at this point, but let’s follow what we just created. Note the new tilde allows another inference. Think and then hit the right arrow key to continue.


Billings BillingsSan Fran ~ ~ San FranL.A. ~ ~ ~ L.A.Tucson ~ ~ ~ ~ Tucson



We get a new ‘X’ from the three on the same row tildes, This ‘X’ allows us to fill in more tildes. Let’s do that. Hit the right arrow key to continue.


Billings BillingsSan Fran ~ ~ San FranL.A. ~ ~ ~ L.A.Tucson ~ X ~ ~ ~ Tucson



The L.A./Perry tilde allows us to see who is from L.A. i.e. Larry; Let’s fill in that new X. Hit the right arrow key to continue.


Billings ~ BillingsSan Fran ~ ~ San FranL.A. ~ ~ ~ ~ L.A.Tucson ~ X ~ ~ ~ Tucson



If Larry is from L.A. then he is not from San Fran or Billings (Using principles 1 and 2 for the next step.) Let’s fill in the two new tildes. Hit the right arrow key to continue.


Billings ~ BillingsSan Fran ~ ~ San FranL.A. X ~ ~ ~ ~ L.A.Tucson ~ X ~ ~ ~ Tucson



Again there are many ways to go. Not with principles one and two but with the identity principles. We have 3 X’s that we have not explored with the identity principles yet. What can we infer from the Larry/L.A. ‘X’? Think before you hit the right arrow key to continue.


Billings ~ ~ BillingsSan Fran ~ ~ ~ San FranL.A. X ~ ~ ~ ~ L.A.Tucson ~ X ~ ~ ~ Tucson



What can we infer from the Larry/L.A. ‘X’? We can infer that if Larry is the guy from L.A. and Larry does not have the blue or red backpack then the guy from L.A. doesn’t have an blue or red backpack. Furthermore, since Larry is from L.A. and the guy from L.A. does not have a green backpack then Larry does not have a green backpack. Three more ‘~’s. Think before you hit the right arrow key to continue.


Billings ~ ~ BillingsSan Fran ~ ~ ~ San FranL.A. X ~ ~ ~ ~ L.A.Tucson ~ X ~ ~ ~ Tucson



One of those three tildes allow us to complete the bottom rectangle. We can fill it in using principles 1 and 2 Think before you hit the right arrow key to continue.


Billings ~ ~ BillingsSan Fran ~ ~ ~ San FranL.A. X ~ ~ ~ ~ ~ ~ L.A.Tucson ~ X ~ ~ ~ Tucson

blue ~ ~ X ~orange ~ ~red ~ ~ ~ Xgreen ~ ~ ~


Hit the right arrow key to continue.



blue ~ ~ X ~orange ~ ~red ~ ~ ~ Xgreen ~ X ~ ~





blue ~ ~ X ~orange ~ ~ ~red ~ ~ ~ Xgreen ~ X ~ ~


The color versus hometown rectangle deserves an ‘X’ by principle 1 leading to two ‘~’s by principle 2. Imagine and hit the right arrow key to continue.



blue ~ ~ X ~orange X ~ ~ ~red ~ ~ ~ Xgreen ~ X ~ ~




Billings ~ ~ BillingsSan Fran ~ ~ ~ San FranL.A. X ~ ~ ~ ~ X ~ ~ L.A.Tucson ~ X ~ ~ ~ Tucson



Again it looks like we might want to use principles 3 and 4. What could be useful? How about this: We see Kevin with a red backpack and Kevin is not from Tucson, so the guy with a red backpack is not from Tucson. Let’s put in that ‘~’. Hit the right arrow key to continue.


Billings ~ ~ ~ BillingsSan Fran ~ ~ ~ ~ San FranL.A. X ~ ~ ~ ~ X ~ ~ L.A.Tucson ~ X ~ ~ ~ ~ Tucson



Now it looks like we can fill in the color versus hometown rectangle with principles one and 2. Hit the right arrow key to continue.


Billings ~ ~ ~ BillingsSan Fran ~ ~ ~ ~ San FranL.A. X ~ ~ ~ ~ X ~ ~ L.A.Tucson ~ X ~ ~ ~ ~ ~ Tucson





Billings ~ ~ ~ BillingsSan Fran ~ ~ ~ ~ San FranL.A. X ~ ~ ~ ~ X ~ ~ L.A.Tucson ~ X ~ ~ ~ ~ ~ X Tucson





Billings ~ ~ ~ ~ BillingsSan Fran ~ ~ ~ ~ ~ San FranL.A. X ~ ~ ~ ~ X ~ ~ L.A.Tucson ~ X ~ ~ ~ ~ ~ X Tucson





Billings ~ ~ ~ ~ BillingsSan Fran ~ ~ X ~ ~ ~ San FranL.A. X ~ ~ ~ ~ X ~ ~ L.A.Tucson ~ X ~ ~ ~ ~ ~ X Tucson





Billings ~ ~ ~ ~ ~ BillingsSan Fran ~ ~ X ~ ~ ~ San FranL.A. X ~ ~ ~ ~ X ~ ~ L.A.Tucson ~ X ~ ~ ~ ~ ~ X Tucson





Billings ~ ~ ~ ~ X ~ BillingsSan Fran ~ ~ X ~ ~ ~ San FranL.A. X ~ ~ ~ ~ X ~ ~ L.A.Tucson ~ X ~ ~ ~ ~ ~ X Tucson



Now we can use principle #4. Kevin has a red backpack and the man with the red backpack is from Billings, thus, Kevin is from Billings. Let’s put in that ‘X’. Hit the right arrow key to continue.


Billings ~ ~ ~ ~ X ~ BillingsSan Fran ~ ~ X ~ ~ ~ San FranL.A. X ~ ~ ~ ~ X ~ ~ L.A.Tucson ~ X ~ ~ ~ ~ ~ X Tucson



That ‘X’ allows us to put in two ‘~’, so let’s do that. Hit the right arrow key to continue.


Billings ~ ~ X ~ ~ X ~ BillingsSan Fran ~ ~ X ~ ~ ~ San FranL.A. X ~ ~ ~ ~ X ~ ~ L.A.Tucson ~ X ~ ~ ~ ~ ~ X Tucson



And those tildes tell us about the last square. Hit the right arrow key to continue.


Billings ~ ~ ~ X ~ ~ X ~ BillingsSan Fran ~ ~ ~ X ~ ~ ~ San FranL.A. X ~ ~ ~ ~ X ~ ~ L.A.Tucson ~ X ~ ~ ~ ~ ~ X Tucson



So we have the complete description spelled out in tabular from. It says1: Larry is from L.A. and has an orange backpack. 2: Perry is from Tucson and has a green backpack.3: Fred is from San Fran and has a blue backpack.4: Kevin is from Billings and has a red backpack.

The first statement ‘Larry is from L.A. and has an orange backpack’. Makes 3 identity claims: Larry is from L.A., Larry has an orange backpack and the guy from L.A. has an orange backpack. To check your answer you would want to look in all three of those corresponding squares. You could also check statements 1 and 4 above against the original statement of the puzzle to make sure it is consistent with this answer. Hit the right arrow key to continue.


Billings ~ ~ ~ X ~ ~ X ~ BillingsSan Fran ~ ~ X ~ X ~ ~ ~ San FranL.A. X ~ ~ ~ ~ X ~ ~ L.A.Tucson ~ X ~ ~ ~ ~ ~ X Tucson



That’s all folks. Hit the right arrow key to end.


Billings ~ ~ ~ X ~ ~ X ~ BillingsSan Fran ~ ~ X ~ X ~ ~ ~ San FranL.A. X ~ ~ ~ ~ X ~ ~ L.A.Tucson ~ X ~ ~ ~ ~ ~ X Tucson



This is a slide show to explain in detail how to solve a puzzle of a common sort. Use the right...

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