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From the days I started to learn Maths, I've have been taught that

Adding Odd times Odd numbers the Answer always would be Odd
e.g 3 + 5 + 1 = 9

OK, but look at this question

UPSC Question

This question was solved and the answer was 30, how it was possible? Need a valid explanation please.

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    $\begingroup$ Do you believe the statement that somebody solved it fairly? Why? There is an old riddle that you have three bags and thirty stones. You need to put the stones in the bags so each bag contains an odd number of stones. One solution is to put fifteen stones into each of two bags, then put one of those two bags into the third. That doesn't work here. $\endgroup$ – Ross Millikan Aug 24 '17 at 2:55
  • $\begingroup$ Why -ve 1, please comment $\endgroup$ – Arshad Ali Aug 24 '17 at 3:05
  • $\begingroup$ I don't understand the downvotes, either. Added the recreational-mathematics and puzzle tags where the question may be better received. $\endgroup$ – dxiv Aug 24 '17 at 3:11
  • $\begingroup$ I didn't downvote. I am surprised that this got an upvote. It also got two downvotes (as of this comment). People seem to agree with my question of why you believe the statement that somebody solved it because of the reason you gave. $\endgroup$ – Ross Millikan Aug 24 '17 at 3:12
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    $\begingroup$ It's possible there is a misunderstanding about "XYZ could solve it". It may mean that XYZ correctly answered the question "can you solve this <equation>".. Presumably his solution stated "no, we cannot solve <equation>" for the reason you gave. $\endgroup$ – Bill Dubuque Aug 24 '17 at 3:48
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You have answered your own question: the sum of an odd number of odd numbers must be odd. Therefore it cannot equal 30.

You should not believe everything you read in a photo on the internet.

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  • $\begingroup$ That's so kind of you! $\endgroup$ – Arshad Ali Aug 24 '17 at 3:00
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Clearly impossible in base 10. Can it be done in another number base?

In base 5: $11+11+3=30$

Actually there are many other possibilities!

Another possibility: fill the boxes with $\binom{5}{3}$,15 and 5.

It holds that $\binom{5}{3}+15+5=30$ (base 10). Notice that $\binom{5}{3}=10$. I can't see any rule being violated as I'm using the 2 parenthesis in the list of valid symbols provided in addition to the numbers 5, 3, 15 and 5 and no extra symbol.

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  • $\begingroup$ May be! this one makes sense a lot. $\endgroup$ – Arshad Ali Aug 24 '17 at 17:56
  • $\begingroup$ If you do $1+11+13$ you won't even need to use a number twice. $\endgroup$ – Jyrki Lahtonen Aug 24 '17 at 18:21
  • $\begingroup$ Or 1+11+15=30 (base 7), just to have another instance without repetition, in another base :). $\endgroup$ – bluemaster Aug 24 '17 at 23:52
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Do you have to fill in all three boxes? I would just put 15 in two of the boxes, e.g., $$\fbox{ } + \fbox{15} + \fbox{15} = 30,$$ and hope people understand the first box as being an implicit 0.

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If this is a riddle, I would do : 13,1 + 7,9 + 9

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  • $\begingroup$ You didn't like my answer? What about this: (1+3)+19+11 or (1+1)+13+15? These questions are usually asked to check if people think out of the box or not. $\endgroup$ – cgiovanardi Aug 24 '17 at 18:28
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    $\begingroup$ Imitation is the sincerest form of flattery, as the saying goes ;-) $\endgroup$ – dxiv Aug 26 '17 at 0:35
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you can also repeat the numbers

Wonder if that means $\,11,5+13,5+5=30\,$ (where the $\,,\,$ comma works as decimal separator).

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  • $\begingroup$ Ooh! so Strange... $\endgroup$ – Arshad Ali Aug 24 '17 at 3:02
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    $\begingroup$ @ArshadAli: this is the kind of unfair solution I was talking about. I enjoy puzzles, as in the riddle I wrote in my first comment. They are not fair at a final exam. $\endgroup$ – Ross Millikan Aug 24 '17 at 3:14
  • $\begingroup$ @RossMillikan Completely agree with your comment. That said, I have no idea what a "UPSC final exam" might be. The first page of google hits brings up mainly links to this very puzzle, which is a bit odd in itself. $\endgroup$ – dxiv Aug 24 '17 at 3:19
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    $\begingroup$ Most likely, UPSC = Union Public Service Commission, notorious for their difficult entrance exams for public employees in India. Check here for some previous exams. $\endgroup$ – bluemaster Aug 25 '17 at 7:50
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What about 6+9+15=30?

they don't stay you have to fill the boxes with the numbers in their usual orientation, after all. :-)

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  • Change the base: $$11_9 + 11_9 + 11_9 = 10+10+10 = 30 \text{.}$$

  • Parentheses are in the list of usable box contents. Commas too. But you're upper limited to one pair of parens and to seven commas. $$(15+15,15)+15 = 30 \\ (15+15,15+15) = 30 \text{.}$$ Here, the parens represent the GCD.

Edit:

Change of base can also be made to work if number repetition were eliminated. $$ 11_{5} + 15_{7} + 13_{9} = 6 + 12 + 12 = 30 \text{.} $$

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  • $\begingroup$ parentheses and commas are in the list for usable symbols, but not the + sign. Your second solution uses an extra plus sign inside the first box, not in the list of allowed symbols :). $\endgroup$ – bluemaster Aug 25 '17 at 7:29
  • $\begingroup$ @bluemaster : Meh. It was late enough I didn't notice. But, ... [edited] $\endgroup$ – Eric Towers Aug 25 '17 at 13:27
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It's $$3_3+3_3+3_3=30.$$ Where $3_3$ is read as 3 base 3.

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