# Unorthodox angle measuring device

I'm using a certain device to measure angles. These are the results I'm getting:

90°  + 90°  = 150°
210° + 60°  = 240°
330° + 150° = 150°
180° + 150° = 300°
240° - 120° = 150°
240° - 180° = 90°
60°  + 240° = 270°
300° - 330° = 300°


What device am I using?

Hint 1:

The device fits in a 0.5 x 0.5 x 0.5 m box.

Hint 2:

The device involves rotation in its operation.

Hint 3:

The device saw most use in the 20th century.

• Interesting observation: rot13(Gur nafjref ner nyjnlf guvegl qrterrf bss sebz rkcrpgrq…) Commented Sep 1 at 16:13

The device is a:

Rotary phone. In particular, its dial, such as this one here (image from wikipedia):
The angle is measured between the "metal stop" and the holes. Therefore:

   0° = (not measurable)
30° = (not measurable)
60° = 1
90° = 2
120° = 3
150° = 4
180° = 5
210° = 6
240° = 7
270° = 8
300° = 9
330° = 0

Then:

Take the original measurements and substitute angles for digits:

 90°  + 90°  = 150° -> 2 + 2 = 4
210° + 60°  = 240° -> 6 + 1 = 7
330° + 150° = 150° -> 0 + 4 = 4
180° + 150° = 300° -> 5 + 4 = 9
240° - 120° = 150° -> 7 - 3 = 4
240° - 180° =  90° -> 7 - 5 = 2
60°  + 240° = 270° -> 1 + 7 = 8
300° - 330° = 300° -> 9 - 0 = 9

This device also fits the three hints:

1. A rotary phone fits in a 0.5 x 0.5 x 0.5 m box.
2. A rotary phone involves rotation in its operation.
3. A rotary phone saw most use in the 20th century.

• Correct! How did you come to this conclusion? What was the thought process? Commented Sep 7 at 1:11
• @webadventurer My first thought was that it had to do with a clock, because the hour hand would move 30° which would affect the final result. But I couldn't make that work and other people guessed that as well and it was ruled out. I only made progress once I saw the 3rd hint. I started thinking of obsolete devices and once I looked up an image of this one, it all fell into place.
– JS1
Commented Sep 7 at 1:35

Not an exact solution, but worth a try

Could you be using a

thermometer?

How?

You convert the "degrees measurements" from Farhenheit into Celsius degrees, perform the operation, and then convert them back to Farhenheit (by rounding them). Examples:
90°F + 90°F = 32.2°C + 32.2°C = 64.4°C = 147.9°F ~ 150°F
180°F + 150°F = 82.2°C + 65.6°C = 147.8°C = 298°F ~ 300°F
240°F - 120°F = 115.6°C - 48.9°C = 66.7°C = 152°F ~ 150°F

The issue I have is

this works for every line, except 330°+150°=150° and 300°-330°=300°. I don't know if I'm close but missing a piece of the puzzle, or completely off tracks!

• It is NOT a thermometer of any kind. You're off track. Interesting attenpt though. Commented Sep 1 at 10:18

I'm not really sure about the precision/accuracy but I think this could be on the right track. If not the correct method, then at least that it's related to the

Minute hand and hour hand of an analog clock

If we convert the first term in each row of the list

to:

3 o'clock + 90° = 150°
7 o'clock + 60° = 240°
11 o'clock + 150° = 150°
6 o'clock + 150° = 300°
8 o'clock - 120° = 150°
8 o'clock - 180° = 90°
2 o'clock + 240° = 270°
10 o'clock - 330° = 300°

I converted the first terms to

whole hours because it's exactly 90° degrees between the hour hand and the minute hand when it's 3 o'clock, exactly 210° degrees between the hour hand and minute hand when it's 7 o'clock, and so on...

Now in each step, the second term, is an operation where you

first rotate the minute hand x° and then rotate the hour hand x°, clockwise or counterclockwise, depending if it's plus/minus x°. The resulting angle between the two hands is the "sum".

But like I said, I'm not really sure about the accuracy but if you try this with

a clock-angle calculator online it's pretty close.

• Good attempt but it's not the right answer. The device is NOT a clock of any kind. Note that there are no 30° and 0°/360° in the equations. Commented Sep 5 at 16:12
• @webadventurer Okay I was actually wondering why there weren't any there but thought that it might be on the right track anyways. Commented Sep 5 at 16:28
• I was sure it was going to be a clock but couldn't pin down exactly why or how. Glad I wasn't the only one! :)
– Stiv
Commented Sep 5 at 19:33

Don't know such a device, but it's technically possible:

90°  + 90°  = 150°
210° + 60°  = 240°
330° + 150° = 150°
180° + 150° = 300°
240° - 120° = 150°
240° - 180° = 90°
60°  + 240° = 270°
300° - 330° = 300°


Whenever there's an addition, the result is $$30°$$ less than the real sum. The opposite is the case for differences. The only exceptions are:

$$330° + 150° = 150°$$
$$300° - 330° = 300°$$

A protractor like this is possible:

The indigo part measures angles properly and is used to see the result.

The red part is used to add or subtract angles, but doesn't show them properly. It tends to show an angle $$30$$ degrees wider than it actually is, but it's a moving part where the main zero can't be rotated towards the first angle if it's wider than a certain limit, so then you'd have to move the second zero (in parentheses) instead, and use 330 as the full circle in parentheses.

• Albeit far from the correct answer, this answer is on the right track. Commented Sep 2 at 12:17
• Without two exceptions ratchet is ideal tool.
– z100
Commented Sep 5 at 18:09

I have found a set of rules to make all the equations work, but I am yet to find any device that follows them.

We treat a revolution as 300° instead of 360°.
In each equation, we remove 30° from the second angle being added/subtracted.
We represent 0° as 300°.

Using these, we can show:

90°  + 90°  => 90°  + 60°  = 150°
210° + 60°  => 210° + 30°  = 240°
330° + 150° => 330° + 120° = 450° = 150° (mod 300°)
180° + 150° => 180° + 120° = 300°
240° - 120° => 240° - 90°  = 150°
240° - 180° => 240° - 150° = 90°
60°  + 240° => 60°  + 210° = 270°
300° - 330° => 300° - 300° = 0° = 300°

Is it about a geographically measurement device? I'm not sure yet, but I have some geometrically understandings...

90° + 90° = 150°

210° + 60° = 240°

330° + 150° = 150°

180° + 150° = 300°

240° - 120° = 150°

240° - 180° = 90°

60° + 240° = 270°

300° - 330° = 300°

• Interesting guess! But this puzzle is not about geography. Commented Sep 2 at 15:29
• @webadventurer Well... I am looking forward to the final correct answer! :D
– tToE
Commented Sep 2 at 15:33
• What program did you use to render these images? Commented Sep 2 at 16:53
• @webadventurer geogebra, an online calculator
– tToE
Commented Sep 3 at 0:43

Same as SquareFinder. I have a rule but no matching device. Maybe it inspires someone.

The rule is:
- add the numbers modulo 360.
- if the result is between 180 and 360 then add 30 else subtract 30.

Another thing to note: all given numbers are multiples of 30˚. I don't know whether it is relevant.