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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:
1. Everything about this puzzle and the answer are in base ten. No other base is used.
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.

pacoverflow
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  • how does Anthony know the numbers are in base 10? :) – Marius May 09 '16 at 15:07
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    You forgot to mention when he crashes it – Raystafarian May 09 '16 at 15:17
  • Since this does not have the tag lateral-thinking can I assume there is no scenario like "he zooms to far and he does not see the full piece of paper" or "there is something written on the back" ? – Marius May 09 '16 at 15:25
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    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. –  May 09 '16 at 15:56
  • Do they follow proper order of operations? – econoMichael May 09 '16 at 17:24
  • @econoMichael I doubt it given -44x6 = -24x11 and both are 1/2 when /528 – Raystafarian May 09 '16 at 18:16
  • Ok. I had a couple of ideas on how to manipulate the numbers but wasn't sure if I should be following order of operations afterward. Thanks. – econoMichael May 09 '16 at 18:25
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    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 09 '16 at 18:48
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    Stating the numbers are in base 10 is actually no limitation at all. – Sleafar May 09 '16 at 19:05
  • @LuxxMiner x is not a variable. Sleafar: The calculation was done in base 10. – pacoverflow May 09 '16 at 20:45
  • It could be the Drone is too far away and didn't capture decimals, but I can't figure out the equation – Alex May 09 '16 at 21:15
  • My supposition is that the drone camera has something to do with this puzzle - I figure that the camera is mirrored and the equation is written backwards or upside down (or both). It works for the majority of the numbers but I'm not sure how to accommodate the 3 and 4's. Edit: their backs are to the window, which suggests that the paper is oriented correctly, but the image could still be mirrored. – Bulldogg6404 May 09 '16 at 23:13
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    Hmm, not sure, but (344-446)/(528-24*11) = -0.4848484848 which is pretty cool :-) – dlu May 10 '16 at 00:44
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    @GentlePurpleRain Maybe they both just made a slight error. Alternatively, your answer could be correct for very high values of 101 – paste May 10 '16 at 01:49
  • @Bulldogg6404 Maybe upside down the 4->h and 3->E? 11xh2 - 825/9xhh - hxhe = 101 – Trenin May 10 '16 at 15:18
  • Maybe 101 isn't a numeric value but a bitmask somehow representing the three expressions separated by dashes (or minus signs) – paste May 10 '16 at 15:40
  • Well, I checked every possible order of operations and none of them came out to 101, so you don't have to worry about that. – paste May 10 '16 at 21:25
  • pacoverflow, would you like to indicate whether anything in the framing story is in any way helpful for solving the problem, or whether it would be essentially the exact same problem if it just said something like "How can you interpret this equation to make it true? The right-hand side really is pronounced 'one hundred and one'."? – Gareth McCaughan May 10 '16 at 21:39
  • pacoverflow, just to reiterate Sleafar's point more explicitly: saying that the calculations are "in base 10" is completely content-free if the "10" there could be in the same base as the calculations: every base is "base 10". Are you intending to tell us that the calculations are in base ten? – Gareth McCaughan May 10 '16 at 21:45
  • (I don't think other bases could actually help unless there's extra weirdness on top. Whatever the base, 34x4 is smaller than 24x11 and 44x6/528 is between 0 and 1.) – Gareth McCaughan May 10 '16 at 21:47
  • He verified that the orientation is correct, but not that it wasn't mirrored. Might warrant further investigating. – Bulldogg6404 May 10 '16 at 22:13
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    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
  • "a man and a woman are sitting at a table with their backs to the window". the fact that they have their backs to the window (and Anthony does not see their faces) is relevant, or is just there for the sake of the story. – Marius May 11 '16 at 06:33
  • Is this maybe tire sizing? https://www.cokertire.com/sml-combo-31-34x4-tr-135-rs-tube.html – Trenin May 11 '16 at 13:48
  • Or picture frames? 4x6 is a common picture frame size. I just posted a goofy answer that uses picture frames. – Trenin May 11 '16 at 14:06
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    There are 24x11 tyres too (e.g. https://www.amazon.co.uk/Maxxis-TYRE-24x11-10-40J-C828/dp/B008K1CRAC) and for both 34x4 and 24x11 the first search result Google gives me is about tyres. But 44x6 doesn't appear to be a standard tyre size (so far as Google knows; I personally am entirely ignorant). – Gareth McCaughan May 11 '16 at 15:25
  • One of the things I've noticed is the phrasing. It's not "The equation was done in base 10", but rather, "The CALCULATION was done in base ten." So, the answer 101 is base ten (wonder why it's "ten" and not "10"), but the equation itself may not be? Maybe someone can run with that. I thought about 34 base 4 etc, but couldn't get it to work. – Josh May 11 '16 at 16:13
  • @Josh: i think this formulation comes from Sleafar remark: number in base 10 mean nothing in reality. By reaction, OP was edited by explicitly replace number in base 10 by calculation made in base ten. I consequently consider that no base consideration is necessary here to find the answer. – aluriak May 11 '16 at 16:26
  • @aluriak - Possible, after having read the other comments (I forget to do that sometimes). If the base is of no consideration, I'd suggest a change to phrasing of "The equation, as well as the answer, are given in base ten." Just a thought. This is a very good puzzle, one I've been trying to figure out for the past couple days! – Josh May 11 '16 at 16:39
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    At this point, I'm starting to think the correct answer is "no". – paste May 11 '16 at 22:16
  • I'm still waiting for McMagister to login and see this puzzle. I wonder if he can solve this puzzle as quickly as he solved my last puzzle that stumped everyone here. :-) – pacoverflow May 11 '16 at 22:51
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    I feel like whatever "wasn't working" is important to the puzzle, but I can't figure how it would affect what was written down... – paste May 12 '16 at 03:41
  • Maybe parts of the equation are computed using Prefix/Postfix as opposed to Infix. – Wesley Situ May 12 '16 at 22:06
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    @pacoverflow Just to make sure this wasn't a typo or something, the only difference between the first and third equations is the 6 changes to a 9? – paste May 13 '16 at 01:49
  • @paste Correct, that is the only difference. – pacoverflow May 13 '16 at 03:51
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    +1 Sometimes before a puzzle is solved, you can't tell if it's going to be an "Aha!" or a groan when the answer is revealed. This one is definitely an "Aha!" Nice puzzle. – Dan Russell May 13 '16 at 15:12

6 Answers6

45

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 : FirstMessage

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:

enter image description here

Jonathan Allan
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Rauwyn
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18

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.

Long Nguyen
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    +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
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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

Sphinxxx
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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

Trenin
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    An interesting answer, but your explanation for 528 isn't very satisfying. – pacoverflow May 11 '16 at 14:56
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    @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
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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

Trenin
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  • 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
1

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

$round(34\times4-(44+6)/528-(24+11))=101$

Jonathan Allan
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James
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