<— Back to ’Hal’s House of Hats’ Logic Puzzle
While you are reading this walkthrough, remember that every hat costs either $1, $5 or $10. This is important.
1. The baseball cap’s clue tells us, right off the bat, that the sun hat costs $10. But there’s another clue that will be very important later on, so let’s analyze it now.
The baseball cap’s clue tells us that each corner is adjacent to three hats that total $16. Now, what three prices can yield that total?
Firstly, we can conclude that at least one of them has to cost $10. Otherwise, the highest collective price they could reach would be $15 (5 + 5 + 5). But two hats that cost $10 would total $20, which is well over the $16 mark. So of the three hats surrounding a corner, exactly one costs $10.
Hat #1: $10
Hat #2: ???
Hat #3: ???
Hats #2 and #3 have to cost $6 combined to make an overall total of $16. Thus, we’re left with this as the only possible combination of prices:
Hat #1: $10
Hat #2: $5
Hat #3: $1
This means that each corner hat is adjacent to exactly one hat of each possible price. Remember that for later.
2. The tinfoil hat is the opposite corner of the baseball cap, so according to the sun hat’s clue, it must cost $1.
3. The tinfoil hat’s clue tells us that the coonskin cap costs $1.
4. Let’s look at the three hats adjacent to the bowler hat; the fedora, the top hat and the viking helmet. As we previously established with our analysis of the baseball cap’s clue, exactly one of those three hats must cost $10.
- It can’t be the fedora, because the fedora and the sports visor cost only $6 combined (coonskin cap’s clue).
- It can’t be the viking helmet, either. It’s in the fifth row, and the five hats in the fifth row total $13 (tinfoil hat’s clue). If there was even one $10 hat in the fifth row, the combined total would have to be at least $14 (10 + 1 + 1 + 1 + 1).
- Therefore, the top hat must cost $10.
- The same logic can be applied to the three hats surrounding the tinfoil hat. The sports visor can’t cost $10 (coonskin cap’s clue), and the beret can’t either, since it’s in the fifth row (tinfoil hat’s clue). Therefore, the hairnet costs $10.
5. According to the top hat’s clue, the five hats in the fourth row total $27. Let’s look at what we know:
Fedora = ???
Top Hat = $10
Army Helmet = ???
Hairnet = $10
Sports Visor = ???
Now, while we don’t know what the individual prices of the fedora and the sports visor are, we do know that they cost $6 combined (coonskin cap’s clue). So…
Fedora + Sports Visor: $6
Top Hat: $10
Army Helmet: ???
Hairnet: $10
6 + 10 + 10 = 26. The army helmet must cost $1 to make a total of 27.
6. With the army helmet’s clue, we can figure out the prices of the remaining two corner hats.
We already know that the bowler hat and the sombrero must have the same price (sun hat’s clue). But their price can’t be the same as that of the other two corners (army helmet’s clue), so they either both cost $5 or both cost $10.
But the bowler hat can’t cost $10, since it’s in the fifth row (tinfoil hat’s clue). So we’re only left with one possibility: both the bowler hat and the sombrero cost $5.
7. The sombrero’s clue tells us that the yarmulke costs $1.
8. As mentioned before, the five hats in the fifth row cost $13 combined. Let’s take a look at what we know:
Bowler Hat = $5
Viking Helmet = ???
Bonnet = ???
Beret = ???
Tinfoil Hat = $1
Unfortunately, we don’t have the information necessary to go any further than this. Or do we?
Recall that the fedora and the sports visor, combined, cost $6. This means that one of them costs $1 and the other costs $5. Let’s look at the possible outcomes:
- If the fedora costs $1, then the sports visor costs $5. That’s obvious. But this also means that the viking helmet must cost $5 as well, since the three hats adjacent to the bowler hat must equal $16 (baseball cap’s clue). By the same token, the beret must cost $1, since the three hats adjacent to the tinfoil hat must also meet this total. Thus, we would be left with this outcome:
Bowler Hat = $5
Viking Helmet = $5
Bonnet = ???
Beret = $1
Tinfoil Hat = $1
- If the fedora costs $5, then the sports visor would cost $1, as would the viking helmet. The beret would cost $5. This would be the other possible outcome:
Bowler Hat = $5
Viking Helmet = $1
Bonnet = ???
Beret = $5
Tinfoil Hat = $1
Either way, the bonnet would have to cost $1 in order to meet the total of $13.
- And now that we know the bonnet’s price, we can figure out the rest of the hats in the third column. According to the sun hat’s clue, the five hats must cost a total of $32. Here’s what we know:
Bonnet = $1
Army Helmet = $1
Sun Hat = $10
This means that the remaining two hats have to cost $20 combined. That can only be attained if they both have the highest possible price ($10 each). The propeller beanie and the hard hat cost $10.
9. According to the yarmulke’s clue, the sports visor and the party hat cost the same. We already know that the sports visor either costs $1 or $5 (coonskin cap’s clue), but what of the party hat?
As we’ve previously established, the baseball cap is adjacent to exactly one hat of each possible price, since it’s a corner hat. We already know that the coonskin cap costs $1, so between the party hat and the chef’s hat, one is $10 and the other is $5.
So the party hat either costs $5 or $10, and the sports visor either costs $1 or $5. The only possible price they have in common is $5.
Both the party hat and the sports visor cost $5. This, in turn, means that the fedora costs $1, the viking helmet costs $5, and the beret costs $1. Also, the chef’s hat costs $10.
10. The combined price of the five hats in the first row is a prime number (propeller beanie’s clue). Let’s see what we know so far:
Baseball Cap = $1
Party Hat = $5
Propeller Beanie = $10
Ushanka = ???
Sombrero = $5
The four hats other than the ushanka come out to a total of $21. Let’s look at the possible outcomes, depending on the ushanka’s price:
21 + 1 = 22 (not a prime number)
21 + 5 = 26 (not a prime number)
21 + 10 = 31 (a prime number)
The ushanka costs $10.
11. Between the jester hat and the beer helmet, one costs $5 and the other costs $10. However, there are already two $5 hats in the jester hat’s column (the sombrero and the sports visor), and each column contains a maximum of two $5 hats (sombrero’s clue). The jester hat costs $1, which, in turn, means that the beer helmet costs $5.
12. According to the hard hat’s clue, the price total of the five hats in the first column must equal the price total of the five hats in the fifth column. Once again, let’s look at what we know:
Column 1
Baseball Cap = $1
Chef’s Hat = $10
Cowboy Hat = ???
Fedora = $1
Bowler Hat = $5
TOTAL: $17 + (price of Cowboy Hat)
Column 5
Sombrero = $5
Jester Hat = $1
Pirate Hat = ???
Sports Visor = $5
Tinfoil Hat = $1
TOTAL: $12 + (price of Pirate Hat)
The pirate hat can’t cost $5 (sombrero’s clue), and it can’t cost $1, since that would leave the fifth column’s price total well under that of the first column. Thus, the pirate hat costs $10, bringing the whole column to a sum of $22. In order to match this sum, the cowboy hat must cost $5.
13. According to the bowler hat’s clue, there are nine $10 hats in total. Thus far, we’ve found eight (the propeller beanie, the ushanka, the chef’s hat, the hard hat, the sun hat, the pirate hat, the top hat and the hairnet). The ninth one must be the last remaining unidentified hat… the witch’s hat.