Result of the Day 2 [Problem B] Challenge of my July 2021 Math mini-contest for D.Buzz

in STEMGeeks4 months ago

Day 2 Problem B

Math mini-contest for D.Buzz for July 2021

The decimal number 16,954.248 converts to 1,244.650 in another number system. What is the base of the other number system?

The problem has originally been posted at https://stemgeeks.net/@savvyplayer/julymathday2b.

Answer

≈ 24.9874

24.987 or 24.99 shall also be accepted as correct answers.

Background

To convert any decimal number to any other number system, we use the target number system's base as the divisor of the decimal number. The quotient will be used as the dividend of the succeeding division operation whose divisor remains as the same until the quotient reaches 0. The remainders comprise each of the digits of the target number system in reverse order.

For example, to convert 753 base 10 (which is the decimal number system) to base 8 (which is called the octal number system), we first divide 753 by 8, whose quotient is 94 remainder 1. We then divide 94 by 8, whose quotient is 11 remainder 6. We then divide 11 by 8, which gives 1 remainder 3. We divide 1 by 8, which gives 0 remainder 1. Using the remainders in reverse as the digits of the target octal number system, the result is 1,361 base 8.

However, this problem is slightly different, as we need to get the divisor of the decimal number used to get the resulting number in the target number system.

Solution

Our problem can be represented by 16,954.248 base 10 = 1,244.650 base n, where n is the base of the target number system.

Brute force

This solution is "trial and error" (or "guess and check"), starting from n = 11 and incrementing upwards by 1.

  • If n = 11, 16954/11 = 1541 with a remainder 3. Since 3 is not the last digit of the number (which is 4) of the target number system, we immediately know that 11 is not the base of the target number system.
  • If n = 12, 16954/12 = 1412 remainder 10. Since 10 is not equal to 4, we continue on.
  • If n = 13, 16954/13 = 1304 rem 4. Since the remainder here is 4, we try to get the next remainder. 1304/13 = 100 remainder 4. Since the remainder is 4 again, we try to get the succeeding remainder. 100/13 = 7 remainder 9. Since the remainder of 9 here does not comprise the target number's third digit from the right before the point, we know that n is not 13.
  • If n = 14, 16954/14 = 1211 without remainder.
  • If n = 15, 16954/15 = 1130 r 4. 1130/15 = 75 r 5.
  • If n = 16, 16954/16 = 1059 r 10.
  • If n = 17, 16954/17 = 997 r 5.
  • If n = 18, 16954/18 = 941 r 16.
  • If n = 19, 16954/19 = 892 r 6.
  • If n = 20, 16954/20 = 847 r 14.
  • If n = 21, 16954/21 = 807 r 7.
  • If n = 22, 16954/22 = 770 r 14.
  • If n = 23, 16954/23 = 737 r 3.
  • If n = 24, 16954/24 = 706 r 10.
  • If n = 25, 16954/25 = 678 r 4. 678/25 = 27 r 3. 27/25 = 1 r 2. 1/25 = 0 r 1. The resulting number is 1234 (but the original problem has 1244.65 as the number in the target number system).
  • If n = 26, 16954/26 = 652 r 2.

Since n having a value of 25 brings us so close (yet not exactly) to the number in the target number system, it is safe to assume that the base of the number system is not a whole number.

Algebraic

Because n serves as the target number system, it can be used to determine the base of the target number system to get the origin (decimal) number system. (Refer to the "Background" section above on this post.)

(1) * n³ + (2) * n² + (4) * n + (4) = 16954
n³ + 2n² + 4n - 16950 = 0

Note that this algebraic solution did not include the decimal value, because of two reasons listed below.

  1. Even very powerful calculators won't be able to solve for the value of n if we use the more precise equation of (1) * n³ + (2) * n² + (4) * n + (4) + (6) / n + (5) / n² = 16954.248 (which includes the fractional part).
  2. The error margin is so small (from rounding off the result) that it would not be worth the effort.

We now have a cubic equation in one variable. Solving for the value of n (which is another topic in Math), we get approximately 24.9874, which is our final answer.

Winner: none

1 HIVE has been transferred to the prize pool, which will be awarded to the participant with the highest number of correctly-answered problems after all the challenge problems in this Math mini-contest have been concluded.

Problems like these are much easier to answer (by guessing or analysis) than for its (detailed) solution to be shown.

Special winner: @lolxsbudoy

Although @lolxsbudoy did not get the exact answer which is relatively and absolutely very close to his answer of 25 which is approximately 0.05% away from the correct answer, I have awarded 1 HIVE to him for his effort as our special participant who answered the problem almost correctly. I would like to optionally invite him (again) to show his solution for the problem.

Award transaction at https://hiveblocks.com/tx/d82860df298819b4901f29e09ceacbd3e0627183.

Mentions: @jfang003, @holovision, @eturnerx (@eturnerx-dbuzz), @ahmadmanga (@ahmadmangazap), @appukuttan66, @paultactico2, @dkmathstats, and @minus-pi
Special mentions: @dbuzz, @chrisrice, @jancharlest, and @mehmetfix

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Thank you Savvy! I searched it up in the net about the process of converting decimals to other number systems. I learned these processes, the same with what you showed. I am very happy to learn new things, however it was very hard to solve especially the trial and errors since the base is not determined. I got 25 for the answer because, I am stocked and it was the nearest possible.

stocked

Did you mean "stuck"?

Anyway, I really intended for the answer to be exactly 25, but I accidentally typed "1244.65" instead of "1234.65".

You can celebrate with a !PIZZA with !LUV.

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@lolxsbudoy! I sent you a slice of $PIZZA on behalf of @savvyplayer.

Learn more about $PIZZA Token at hive.pizza (3/10)

<><

<><

LUV

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@lolxsbudoy, you've been given LUV from @savvyplayer.

Check the LUV in your H-E wallet. (3/10)

Thought so hahha.

Yes, I meant stuck... but the auto spell got me.

Thanks again for hosting this contest savvy, I learned new lessons.

It's really good to know that you learned something from my hosted event.

If only there are at least 3 people participating for each challenge problem of my Math mini-contests on Hive (particularly D.Buzz), then I would have hosted Math contests on Hive every single month.

!PIZZA !LUV

<><

<><

LUV

Connect

Trade


@lolxsbudoy, you've been given LUV from @savvyplayer.

Check the LUV in your H-E wallet. (4/10)

Connect

Trade


@lolxsbudoy! I sent you a slice of $PIZZA on behalf of @savvyplayer.

Learn more about $PIZZA Token at hive.pizza (4/10)

Yeah, I understand how difficult creating math problems is and the pain of being somewhat "ignored". Anyways, I understand where you stand savvy. I will look forward to other things you will host, and more new learnings to come hahha.

I notice that there are so much content creators on Hive, but relatively few leave comments on the others' posts or comments.

I also learned that the best way to get engagement on your posts is to actually give away something, like Hive Engine tokens such as !PIZZA and !LUV. I still see you joining #Splinterlands card giveaways, but I no longer have extra cards to send.

<><

<><

LUV

Connect

Trade


@lolxsbudoy, you've been given LUV from @savvyplayer.

Check the LUV in your H-E wallet. (5/10)

Connect

Trade


@lolxsbudoy! I sent you a slice of $PIZZA on behalf of @savvyplayer.

Learn more about $PIZZA Token at hive.pizza (5/10)

Yeah, it is true... there is no much engagement for now except for giveaways and such.

Always thankful for your thoughts of giving me some extra cards, and no worries about that. I also applied your tip about buying common cards and renting the epic and legendary ones. For now, I already have a level 6 common card which is pretty great.

I only have one Level 2 Summoner card, and that is Mylor Crowling which is my favorite. I have some Level 2 Monster cards, though I can't use their upgrades unless I use Mylor Crowling as their Summoner.

Anyway, I have already bought the Legendary Monster Cards The Kraken, Ancient Lich, Dark Ha'on, and High Priest Darius before the Card and DEC prices went through the roof. I won the Gloridax Magus and Spirit Druid Grog from Reward Chests. For obvious reasons, they are all at Level 1.