# Can you explain these equations?

Anthony just got a new drone with an HD camera and microphone, and he is eager to test it out. He connects his computer to its wireless live video stream, opens up his window, and flies it outside. On the video stream, Anthony can see his neighbor's house. He notices that one of their windows is open and a man and a woman are sitting at a table with their backs to the window. He decides to spy on them, so he moves the drone close to the window and has it hover there. The man and woman are talking about something as Anthony watches and listens to them on the video stream.

The man says "... wasn't working, so I had to write everything down and do the calculations myself." Then he hands a piece of paper to the woman and asks her, "Can you double check my math?" As the woman holds the piece of paper, it is entirely visible to the drone's camera, so Anthony zooms in on it. He sees the following equations, and he is really impressed by the man's immaculate and unslanted penmanship:

34x4-44x6/528-24x11 = 101

21x21x21x5/315-62 = 85

34x4-44x9/528-24x11 = 101

The woman looks at the first equation for about 30 seconds and then says "The first one's right, one hundred and one." Then she looks at the second equation for about 30 seconds and says "The second one's right, eighty-five." Then she looks at the third equation for about 30 seconds and says "The third one's right, one hundred and one." The man says "Thanks", and then starts to get up from the table. Anthony quickly flies the drone back to his house.

After downloading the video recorded by the drone to his computer, Anthony plays it back. He looks at the equations again, and he can't figure out how the man and woman did the calculations. Can you explain how they did the calculations?

Hints:
2. Anthony saw everything that was written down, and he saw it in the correct orientation without mirroring. The man and woman did not do anything that Anthony couldn't see or hear.
3. The man and woman performed the calculations correctly.
4. All of the equations use the same calculation method.

• Please clarify whether the "x"s in the equation are multiplication signs or variables. If it's just a variable, I think the equation would be correct when x = -(202/257) (I don't believe that's the answer you want). But if they are multiplication signs, maybe a "*" instead of a "x" would be better. – user14478 May 9 '16 at 15:56
• If you assume there are parentheses around each pair of numbers, starting at the end, you get $34\times(4-(44\times(6\div(528-(24\times11))))) = 102$, which is close... – GentlePurpleRain May 9 '16 at 18:48
• Stating the numbers are in base 10 is actually no limitation at all. – Sleafar May 9 '16 at 19:05
• Also, just in case anyone here hasn't checked the revisions, there is a slight hint there in when OP reverted someone else's edit. – paste May 10 '16 at 23:47
• At this point, I'm starting to think the correct answer is "no". – paste May 11 '16 at 22:16

The equations are

bowling scores. The symbols don't represent math operations - "X" is a strike, "/" is a spare, and "-" is no score.

I don't really feel like calculating all of them, but for the first and second one:

First message :
Second message
First frame is 2, then 1 for a score of 3
Second frame is a strike
Third frame is 2, then 1, that plus the strike gives 3 + 13 + 3 = 19
Fourth frame is another strike
Fifth frame is 2, then 1, that plus the strike gives 19 + 13 + 3 = 35
Sixth frame is another strike
Seventh frame is 5 then spare, the sixth frame score is then 20
Eighth frame is 3, then 1, the spare then gave 13 points for a current score of 35 + 20 + 13 + 4 = 72
Ninth frame is 5, then nothing, 72+5=77
Tenth frame is 6, then 2, 77+8=85

The edit history hint helped a lot with this one.

Edit: For completeness here are all three:

• Nice! Given my love for this activity, I'm a bit ashamed I didn't see it. – Dan Russell May 13 '16 at 14:50
• Congratulations on figuring this one out! – pacoverflow May 13 '16 at 15:14
• Added all three, hope AOK. – Jonathan Allan May 13 '16 at 23:45

I don't know if this is just a coincidence or not, but if you:

transform the equation into words

you get:

Thirty four times four minus forty four times six divided by five hundred and twenty eight minus twenty four times eleven, and not counting the spaces this expression has a total of 101 characters.

• +1 for finding a neat coincidence. If this were the answer though, I don't think the puzzle would deserve the [calculation-puzzle] tag. – pacoverflow May 11 '16 at 18:50

On the video, maybe it is...

..hard to tell + and x apart, and also hard to see decimal points and parentheses?

The equation is actually

.34 + .4 - 44 * (6/528 - 2.4 + .11)
which is 101

• Sweet. If that's not right, seems like it's at least a plausible alternate answer. – Dan Russell May 10 '16 at 2:32
• Hint #2 says that Anthony saw everything that was written down, which would include decimal points. – pacoverflow May 10 '16 at 4:08

Turns out the people were actually

ordering picture frames

They needed

3 4x4 sized frames, 4 4x6 sized frames (which they are getting from 528 photography, by the way) and 2 4x11 sized frames.

The cost for these was

4x4 cost \$5, 4x6 cost \$10, and 4x11 cost \$23. Thus 3x5 + 4x10 + 2x23 = 15+40+46=101 • An interesting answer, but your explanation for 528 isn't very satisfying. – pacoverflow May 11 '16 at 14:56 • @pacoverflow Agreed. And there are many solutions to the equation with 3 unknowns. I only provided one with nice round numbers (except for 23, which is my favourite). I don't seriously think this is the answer, but thought I would throw it out there anyway. – Trenin May 11 '16 at 14:58 There are 3 expressions and three digits in the answer. The answer is actually interpreted as 1-0-1 where each digit corresponds to each expression. The meaning is If the expression is greater than 1. Thus 34x4 is greater than 1 44x6/528 is not greater than 1 24x11 is greater than 1 Thus the answer is True-False-True Or equivalently 101 • That's essentially what I said in my comment above, but it seems the recent edit has told us we're incorrect. – paste May 10 '16 at 20:02 • I had the same logic, this could be explain also has is decimal or not, is even or not – AxelH May 12 '16 at 11:22 Maybe his handwriting slants, some 'x' signs read as plus signs And the quantities involved imply brackets around the summed terms and a rounding of the result Yielding$round(34\times4-(44+6)/528-(24+11))=101\$

• Downvoter - when a new user posts it may be worth considering a comment to guide them rather than an instant downvote. @James I improved the post, hope AOK – Jonathan Allan May 12 '16 at 3:26
• Maybe the thing that the man said wasn't working was handwriting recognition software. – paste May 12 '16 at 3:48
• Just seen the answer. Really great puzzle! :-) – James May 13 '16 at 23:12