Thanks to those of you who took part in the little maths problem last week. It was interesting to see the variety of answers coming out. I’m not sure how the first 3 were calculated, but I will try to explain some of the other ones…

**1) 150**

**2) 180**

**3) 20**

**4) 152** because the units have a pattern 8, 2, 0, 2 and tens goes up sequentially by 2 – 1, 3, 5, 7. This give 8 = 112, 9 = 130, 10 = 152

**5) 116** because 3 + 7 = 10, therefore 10 = 18 + 98 = 116

**6) 100** because 5 + 5 = 10 therefore 10 = 50 + 50 = 100

7) 104 because 4 + 6 = 10 therefor 10 = 32 + 72 = 104

**8) 10** clearly 10 = 10

**9) And the most popular (so clearly wrong?) was 200 **because the pattern is 10 * 10 * 2 = 200

UPDATED 25/01/2014 due to additional possibilities:

**10) 62**. To get this, you need to remove the equals sign and calculate the difference between the numbers – this difference increases by 4 between each pair of numbers. So, 432 – 318 = 114, 550 – 432 = 118, 672 – 550 = 122, 798 – 672 = 126. The next number will be 798 + 130 = 928, 928 + 134 = 1062. Now reinstate the = to give 10=62. Phew. Thanks to Tony Bialorucki for this answer.

**11) ?**. Yes question mark. If you read it as a set of statements rather than a question, then 10=?. Thanks to sik for this answer.

There are several issues with how this problem is written, which means you could end up with almost any answer given the correct logic –

First, it starts off with a statistic (99.9%) which is to entice you attempt an answer.

Second, the question itself is not correctly worded. It should go along the lines of ‘Complete the following sequence…’

Thirdly, the calculations are clearly all incorrect, so you could almost put any number down. The logic is deliberately vague

When I originally did this problem, my first answer was 160. This was because I didn’t read the question properly (a problem my 9 year old often has), and assumed it was asking for the next number in the sequence – 8 = 160!

**So, what is the answer?**

It’s 200, although I did like the 10 from The Penguin as this is the only answer which is true!

Sorry if anyone was disappointed, but hope you enjoyed the problem.

Like this post? Then check out more great maths books on Amazon

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How about this math?

The old ones are always the wisest.

How did you get to 8 = 160?

When I saw the problem I found 200 as the answer for 10 = ?

I found it by dividing the given equal by the initial number and adding 2 to my result.

Hence…

3 = 18 ~ 18 divided by 3 gave me 6 and adding 2 made 8; and next in sequence would be (4X8) = 32 and continuing, then…

4 = 32 ~ 32 divided by 4 gave 8 and adding 2 made 10; therefore, (5 X 10) = 50 etc. etc.

5 = 50 ~ (10 + 2) is 12 and (6X12) = 72

6 = 72 ~ (12 +2) is 14 and (7X14) = 98

7 = 98 ~ (14 +2) is 16 and (8X16) = 128

8 = 128 ~ (16 + 2) is 18 and (9X18) = 162

9 = 162 ~ (18 + 2) is 20 and finally (10 X 20) = 200

10 = 200

If my math is flawed I’m sure you will tell me, but I can’t figure out how you found that 8 would be 160 in this algorithm.

Hi Lynda, thanks for trying this out. I got 160 by completely misreading the question – I calculated 10 x 8 x 2 = 160! Basic error I’m afraid, guess it happens with age!

Danny, you have not corrected my math, so I can assume that I was correct in my thinking on the solve.

TRUE CONFESSION: I grew up a mathaphobe and didn’t get over it till I went back to school to become a teacher. I guess when you want something bad enough you will put your head into the lion’s mouth to get it? I am still a bit shaky even to this day, but became confident enough in my training to teach the early primary grades.

Thank you for the fun math task, and for keeping me one step ahead of Alzheimers. 😉

Yes, your logic is correct and 200 is the answer, although there are other answers that also fit if you follow different rules.

Great to hear you overcame your mathaphobia. You’re never to old to learn something, and great you put it to good use too.

I got there slightly off scew, the number on the right increases by 4 at each point, 14,18,22,26,30,34,38, therefore at 8 =128, 9=162 and 10=200!!

Yes, I have seen this method from some other readers before. Maybe I’ll add it as another possible answer.

Thanks for dropping by.

I came up with 10 = 62. If you remove the “=” sign, the difference between each number goes up by 4: 432 – 318 = 114; 550 – 432 = 118; 672 – 550 = 122; 728 – 672 = 126. Following this logic, the next two numbers in the sequence would be 928 (728+130) and 1060 (928 +134). Insert your “=” for 10=64.

Is this what you mean by many solutions, or is there something wrong with my logic?

Interesting answer and a good one I think. Yes, there are many answers for the question, it just depends on the logic you use.

There being no mention of the results being reached thru mulitplication, you must assume only the relationship of the numbers themselves. Therefore, determining the difference = 4 in each when subtracting each succesive result, hte answer must be 38

7 98 26

8 30

9 34

10 38

7 98 26

8 30

9 34

10 38

Hi Eddie,

I can see your line of thinking, but I think your logic doesn’t quite follow –

3=18

4=32 because 32-18 = 14 and 18+14=32

5=50 because 50-32 = 18 and 32+18=50

6=72 because 72-50 = 22 and 50+22=72

7=98 because 98-72 = 26 and 72+26=98

So far so good.

Following the same pattern:

8=98+30=128 and 128-98=30

9=128+34=162 and 162-128=34

10=162+38=200 and 200-162=38

So I still get 200!

Is this what you mean, or did I misunderstand something?

Thanks for dropping by…

This is how I figured the answer of 200.

The diff between 18 & 32 =14, diff between 32 & 50 =18, diff between 50 & 72 =22 and so on…

This shows a sequence of the number increasing by 4 each time…

If the diff between 72 & 98 is 26 then just add another 4 as per the sequence to give you 30

So then add the 30 to 98 to give you the value of (8) which is 128;

Add 34 to give you the value of (9) which is 162

And so finally adding 38 to 162 gives you the value of (10) which is 200

Thanks Graham. Same answer, different route. Just shows how many ways there are to answer this one.

10 equals question mark…there is no instruction to solve any math or complete a sequence in other words it is exactly what is written.

Another good answer. Well done.

160 is a valid answer I think. Here’s the pattern: 3 * 6 = 16 take the multiplier (6) add 2 and multiply by the next number 4 * 8 = 32 5 * 10 (8 +2 = 10) = 50 , 6 * 12 (10+2 = 12) = 72, 7 * 14(12+2 = 14) = 98, 10 * 16 = 160 (14+2 = 16)

I think you made the same mistake I did – going straight from 7=98 > 10=160, missing 8 and 9. Assuming these also form part of the sequence.

Danny – you did not miss part of the sequence. 3,4,5,6,7,10 is in fact the sequence. That is called “3 times the prime sequence”. That sequence continues as 10, 11, 12, 13, 16, 17, 18, 19, 22, 23, 28, 29, 30, 31, 36, 37, 40, 41, 42, 43, 46, 47, 52, 53, 58, 59, 60, 61, 66, 67, 70, 71, 72, 73, 78, 79, 82, 83, 88, 89, 96, 97, 100, 101, 102, 103, 106, 107, 108, 109, 112, 113, 126, 127, 130, 131, 136, 137, 138, 139

Thanks Steve, something to think about I guess.

I got 200 for a different reason LOL. The sequence starts at 14 and adds 4 to iteration.

Reason is, 32-18 = 14 , 50-32 = 18, 72-50 =22 98-72 = 26 so therefore 8 = 128, 9 = 162 and 10 = 200.

Well done, I have seen a few people with the same thinking.

Thanks for dropping by.

n => 2n ^ 2

Just to make it clear – n => 2 (n ^ 2). Looks good to me.

In my humble opinion, the correct answer is: “What?”

99.9% failed because they didn’t understand the question. You can never answer a question if you don’t understand what is being asked. Assuming that you do understand it and basing your answer on that assumption will always lead to an incorrect answer, even if that answer can somehow be “proven” to be correct…

Agreed, I alluded to that in the intro to the ‘answer’. A badly worded question – if it even is a question – gives rise to many answers.

result:=0; x:=10; for(n:=0; n<=(x-1); n++) { result:=result+(2+n*4); } …. in this case 200. [after read all answers for sure the better solution is 2*x^2. I focused on sequence approach.. and if we know the previous step – the next one will be +(2+(x-1)*4) where x is the next step left side of equation]. Very good exercise :]

Thanks for your contribution. Yet another method!

n = 2n^2 (HENCE 200 answer) is as much true as n = 10(2n-5) + 2*[n%4 + (-1)^[(n+1)/2]] (HENCE 152 answer).

Thanks for your contribution. I’m sure a mathematician could prove that there are an infinite number of formulae that fits this puzzle…

reduced… 2 x (n x n)

Exactly.

I answered 200 using the logic that the multipler increases by 2 eachtime. However, my son answered 202 and proved that his answer works, as well. Do you want to try to figure out his method or should I go ahead and post it?

We found a calculation error in his answer so it does come out to 200. His method is still unique, though:

3 = 18

4 = 32 (32-18 = up 14)

5 = 50 (50-32 = up 18 = up 4)

6 = 72 (72-50 = up 22 = up 4)

7 = 98 (98-72 = up 26 = up 4)

8 = 128 (128-98 = up 30 = up 4)

9 = 162 (162-128 =up 34 = up 4)

10 = 200 (200-162 = up 38 = up 4)

Hi Christine, thanks for your input. I think 200 is the most common answer, but as you can see above, there are many methods to getting the same answer!

I have seen as similar answer to this – see Eddie’s comments above (although I think he also made a calculation error) from Jan 22.

f(x) = 2n^2

That’s it. I guess I should put that in the answer for (9) above…

Thanks for dropping by and contributing.

The correct answer is 172.

The pattern: 8, 2, 0, 2, 8…goes on to 2, 0, 2 etc

The first part of the number are all prime numbers. (1, 3, 5, 7, 9…11, 17)

😀

just kidding

OR AM I?

Hi, thanks for your response. This is similar to answer (4) I had above, except you are taking prime numbers rather than just odd numbers.

You have made a mistake though – 1 and 9 aren’t prime numbers!

Good try though.

Dangit!!!! lol

200 is the answer

3 X 6 = 18

4 X 8 = 32

5 X 10 = 50

6 X 12 = 72

7 X 14 = 98

8 X 16 = 128

9 X 18 = 168

10 X 20 = {[(/200\)]}

ANSWER

The answer is 200

I will tell you how

If 3=18

Then it is 3×6=18

If 4=32

Then it is 4×8=32

If it is 5=50

Then it is 5×10=50

So you see I am multiplying the already given number with another number to get the result and I increase it by 2 as I progress

So it is 10 for 5 and then it will be 12 for 6

Which will give 72

And 14 for 7 which will give

14×7=98

And 16 for 8, 18 for 9 and 20 for 10

Ultimately giving 200

I answered that 10 = “?” or if nothing else, 10=10. I didn’t read into the problem that there was ever a question being asked, as I was just assuming that at some point it would ask me to solve for “?”. The fact that there is a pattern in the numbers preceeding the final statement seemed coincidental, as I felt that since it wasn’t establishing “8=128” or “9=162”, I could argue that 8=Tuesday, or some other such random factoid.

I did conclude that if we lived in a world where 5=50, that I needed to get to a bank quickly to exchange some currency.

2(x)^2

2(10)^2=200

Sounds about right. Thanks for visiting

I will be assuming they are using the relation operator for assignment rather than equality, as the former has more interesting consequences.

With a little effort we can come up with say,

(-(1/30))n^5+(5/6)n^4-(49/6)n^3+(247/6)n^2-(459/5)n+84 which results in,

18, 32, 50, 72, 98, 124, 138, 116, . . . .

——-

But,

——-

2n^2 gives

18, 32, 50, 72, 98, 128, 162, 200, . . . .

(The popular choice)

——

However,

—–

n^5-25n^4+245n^3-1173n^2+2754n-2520 gives

18, 32, 50, 72, 98, 248, 882, 2720, . . . .

—–

But then there is,

—–

(8/5)n^5-40n^4+392n^3-1878n^2+(22032/5)n-4032 that gives

18, 32, 50, 72, 98, 320, 1314, 4232, . . . .

—–

And then there is ….

… I can keep going all day, but I’ll just stop there.

—–

All these work, take your pick 🙂

—–

Finally, the point also is that given any finite set of numbers, the set does not necessarily define any single sequence.

I got the correct answer of 200 using this simple method that took no time and was easy:

I took the difference between each number (diff between 18 and 32 = 14, diff between 32 and 50 = 18, diff between 50 and 72 = 22 and so on). Then I took the difference between the new numbers I just came up with (diff between 14 and 18 = 4, diff between 18 and 22 = 4 and so on). All of the new differences are 4 so using the number 4, I was able to back into the missing numbers (8 = 128, 9 = 162 and finally 10 = 200).

Another interesting way of getting the answer of 200! Thanks for the contribution.

x6 x8 x10 x12 x14 x16

10=160

Thanks Matias, it seems to never end!

3=18 (3×6=18)

4=32(4×8=32)

5=50(5×10=50)

6=72(6×12=72)

7=98(7×14=98)

8=128(8×16=128)

9=162(9×18=162)

10=200(10×20=200)

Why wouldn’t it be 130? 3=18.. 4=32 so 32 -18=14..5+50-32=18…6=72-50=12..7=98-72=16 so 8=108-98=10…9=122-108=14..10=130-122=8..Do you see the sequence here????

Hi Sherry, thanks for the attempt. Not sure I quite follow though as you have a couple of calculation errors –

3=18 ok

4=32-18=14 ok

5=50-32=18 ok

6=72-50=22? not 12?

7=98-72=26? not 16?

8=108-98=10 ok

9=122-108=14 ok

10=130-122=8 ok

So I see the two sequences you are trying to create – differences decreasing by two, but 6 and 7 aren’t correct.

Let me know if I’ve misunderstood, otherwise good try!!

Danny