Can You Solve the $$\sqrt {car}=37$$ Puzzle?

[PAllen]

TL;DR

My daughter discovered this using an unusual program I wrote for her.

There is a precise sense in which it is correct to say: $$\sqrt {car}=37$$

The puzzle is to figure out in what sense this is true.

[What made this so exciting for her is that 37 is her favorite number, and that she was obsessed with car models at the time].

 

[256bits]

37 is prime.

but we could have c = 1369
and if a NE r, ( ie if a = r = 1 ), then they can be just the reciprocal of one another.

( I would like the large bunny in the corner for my girlfriend. )

 

[PAllen]

The solution I'm looking for is much more natural than that. I'll give a big hint: $$\sqrt {cdo}=38$$ and $$\sqrt{cpw}=42$$ All 3 square roots have to be true under the same system. One more, just for fun: $$\sqrt{doj}=49$$

 

[Filip Larsen]

Spoiler

Base 26 comes to mind ...

 

[PAllen]

Spoiler

Base 26 comes to mind ...

Winner!

 

[Buzz Bloom]

I think I have solved it. How do I create a "spoiler"?

 

[PAllen]

I think I have solved it. How do I create a "spoiler"?

In edit message window, touch the arrow next to the 3 dots. Spoiler is one of the options.

 

[Buzz Bloom]

Spoiler: This is an incorrect answer

The three letters "c", "a", and "r" each have a different value between 1 and 26 inclusive.
The notation represents a three digit (base 26) number. That is c has a coefficient of 26², a has a coefficient of 26, and r has a coefficient of 1. This produces the equation below.

676 c + 26 a + r = 37² = 1369​

c = 1 because c>1 implies 676 c > 1369.
Therefore,

26 a + r = 693.​

Now

693/26 = 26.65...​

Therefore, a = 26, and r = 693 - 676 = 17.

The cdo problem is also solved the same way, although c now has a different value.
c = 2, 26 d + o = 38² = 1444 - 1352 = 92
Now

92/26 = 3.5...​

Therefore d = 3 and o = 14.

Oops. I now see that I made a mistake. I will correct it in a separate post.

 

[Buzz Bloom]

Spoiler: I think I have it right now

Base on the the cdo solution
c = 2
d = 3
o = 14
in the previous post, I now see that in the car problem
c = 2
a = 1
r = 17.

The general pattern is
each letter has the value of its position in the alphabet - 1.

 

[PAllen]

Spoiler: I think I have it right now

Base on the the cdo solution
c = 2
d = 3
o = 14
in the previous post, I now see that in the car problem
c = 2
a = 1
r = 17.

The general pattern is
each letter has the value of its position in the alphabet - 1.

Correct.