Number sequences and IQ: discussion + problems to solve

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Number sequences and IQ: discussion + problems to solve

Postby Dubravsky » Sat Sep 03, 2011 12:38 pm

Hi

I'm intrigued by what can solving simple (or "simple") numeric sequences (discovering the logic and filling in missing terms) tell about one's intelligence. Years ago I was on a supervised psychological test and there was a sequences subtest of this type. I found the sequences pretty easy and later was told I had hit the ceiling on this. Some time after I stumbled upon some alleged high-range IQ tests on the internet, some of them made purely out of sequences (actually two or three of these were done by people with PhD in psychology). These tests are untimed and unsupervised and what is fascinating about them is the sequences themselves. The solutions (on the good tests) are at the end of the day very simple and fundamental, yet it usually takes a good deal of time to realize what's going on (if you find out at all). Plus it's not like some divine insights, brute force or mathematical knowledge is needed, you just need to follow the clues and deduce. Simply, I was often left in awe and I got strong impression these things can tell a good deal about one's intelligence.

Now I'm no psychologist but I am a student of mathematics and know statistics to analyze data, so I decided to make my own test and to see how people perform and draw conclusions. I have created a good number of what are hopefully interesting and nice puzzles with the potential to tap different reasoning and pattern recognition abilities and measure high-range of mental ability. You may have a look at my test at http://dubravsky.szm.com/seqs1.html . Your opinions on it or just on this topic in general are welcome.

EDIT: I'll probably just add some stuff here, can you solve it? Also post your own puzzles if you want to challenge others. No calculations with big numbers are needed.

1.) 18127, 26328, 34529, 42730, ?

2.) 1001, 2000102, 3012000003, 4000000032104, ?

3.) 7911, 810, 131517, 846, 192123, ?

LD
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Postby tvelection » Tue Sep 06, 2011 12:22 pm

I'll give the others a shot when I have more time, and if no one else has tried, but number 1 might be, . . . 50931?

If the procedure is first number increases by one, second number decreases by even numbers, third number increases by odd numbers, last two numbers increase by one (the last example being 30).
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Postby Dubravsky » Tue Sep 06, 2011 12:45 pm

Yeah that's that, good job :) This one should test the ability to recognize a pattern by decomposing it into constituents, which are elementary.
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Postby tvelection » Wed Sep 07, 2011 12:49 pm

Thanks LD,

I play piano and in doing so (especially with sight reading) deal with recognizing patterns quickly which probably helps with such number pattern problems.

After posting above I thought a mathematician might try to correct me on a technicality in my procedure. They would have asked, "Is zero an even or an odd number??" Meaning I should have said --- increases by 2 and decreases by 2 rather than mention "even" or "odd." But that is a matter of correct terminology since what was meant was clear enough and the answer correct.

I bet creating such sequences is a difficult challenge in itself. It's possible, I suppose, someone may find an unintended patterns that produce an alternate answer than the one intended. Also for the creator to give just enough information to make it solvable but not too much to make it obvious. A good puzzler will usually have an arsenal of analyses via questions. Does it make sense . . . Backwards? By the shape of the numbers? By the alphabetic beginning or end of each number? How many digits does each in the sequence have? Is it related to phone pads? or Clocks (regular or military)? Months? Years? Odds? Evens? The number of letters in the spelling of the numbers? Alphabetic order of spellings? Does it increase or decrease? How are all the first numbers generated? . . . . etc. More generally one may ask, "what is changing and how does it change from entry to entry?" or one may just look at each entry to the next with the least bias and try to see a pattern come out.

Here's a shot at the second one:

5012340000000005 ?

The procedure deals with the number of zeros (in the long string) being equal to the two numbers that sandwich them. All of the entries begin and end with the same number, increasing by one for the first/last number and increase a total 3 added digits for each entry (4, 7, 10, 13, . . . 16). The zeros seem to alternate sides after the first one. The inner count rises at an odd first number and descends at even (or maybe I should say every two). I'm not as sure as with the first one but it seems correct. Don't give me the answer if it is incorrect.
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Postby tvelection » Wed Sep 07, 2011 7:41 pm

May as well try them all. I like how you don't give multiple choice answers because that gives a chance at a blind guess or to see the correct way when the correct answer is seen. I've noticed that in some sequence tests they will leave an earlier entry blank occasionally instead of the last one.

If you have more examples to give I won't participate in case you want others to participate. I like these kinds of problems, especially since my math skills are high school level at best.

last answer: 4113

The procedure is to add the inner two numbers then the two numbers outside them (right and left), and so on, outward to the first and last --or outside inward w/left to right answer. _ (3+1) _ (9+2) _ (2+1).
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Postby Dubravsky » Thu Sep 08, 2011 8:07 am

Concerning music, I have noticed several times musical interests and even accomplishments with gifted people in other areas, look for example at the profile of US international math olympiad team, half of them has also accomplishments in music: http://www.maa.org/news/062609usimo.html

Yes I found your explanation stated in a little strange way (I'd expect the "increase by 2" etc.), but it seemed clear enough you grasped the logic. And after all, only the correct answer is needed. Lastly, zero is definitely an even number.

Concerning sequence creation and solving rules:

1. alternative solutions can be often thought out, but they should be always of such inferior quality that they can barely be considered solutions at all when confronted with the intended ones. This low quality of "alternative solutions" is often marked by:

a. not explaining all of the given data in a consistent way. There are details that don't fit and are explained in an awkward way.

b. The explanation being more complex than the intended one. Both a. and b. often go with the following:

c. Making up rules that you never actually see at work, they are just ad hoc patches to save the day.

Let me give you one example of a bad alternative solution, item 3. (7911, 810, 131517, 846, 192123, ? ) can be explained as:

At odd positions there is a chain of odd numbers beginning with 7, grouped into triplets (7 9 11 | 13 15 17 | 19 21 23); the even positions start with 810 and increase by 36. You never really see a confirmation of the second rule, it's pure fantasy. At this point I note that your solution is correct and you obviously felt that yourself, because you have seen the rule at work two times within the given data, and in fact probably even "a bit more" since the rule is structured into similar subrules. Your other solution is also correct, maybe give it a thought more to get more confident.

These bad alternative solutions are the reason why people always think they will get higher score on a test than they actually do. Finally yes it is still needed to put an effort to check that there isn't some mistake allowing to interpret the given data in more equally good ways.

2. Deducibility - you are right that there should be clues to follow, no divine insights or brute force needed, and of course a variety of difficulty.

3. IMPORTANT - (almost) no acquired knowledge should be needed! Not even something you may think eveyone knows, like the number of days in a month or the number of letters in an English word. This is of utmost importance and all good tests follow this. The only things used are basic arithmetic operations on numbers (and these should be small numbers too, no calculators needed) and knowledge of letters of English alphabet and their ordering.

More sequences:

4.) 1, 000X00000, 9, X0000X000, ?, ?

5.) 1, 112, 112123, ?

6.) 49, -5, 36, -3, 18, ?, ?, 0

If you solve these too you may want to have a look at my test, which is quite challenging I suppose.
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Postby tvelection » Sat Sep 10, 2011 11:22 am

LD wrote:
More sequences:

4.) 1, 000X00000, 9, X0000X000, ?, ?

5.) 1, 112, 112123, ?

6.) 49, -5, 36, -3, 18, ?, ?, 0

If you solve these too you may want to have a look at my test, which is quite challenging I suppose.


I'll have a look at the site/test. I wondered if I should post again at all. So if you'll indulge me once more I'll give these a try too, having waited a few days in case someone else wanted to try.

As for the sequences:

4.) 25, 00X00X000

The procedure involves recognizing that the 0 and X strings represent the numbers 1 to 9 with zeros as place holders and Xs as number markers. this would make the sequence 1, 4, 9, 16, (25, 36) separated by rising odd numbers 3, 5, 7, 9, 11 respectively.

5.) 112123124

Here the key is the first entry (1) has 111 added to it, the next (112) has 11 added. Then the entire answer is added to the end of each consecutive entry. Then in the last entry (112123) 1 is added to give the solution (112123124).


6.) -7, 11

The positive entries 49, 36, 18 produce the others by subtracting the ones column number from the tens column number (4-9) , -5, (3-6), -3, (1-8 ), -7, (1-1), 0. At 18 as the positives descend there is only one 2-digit number left that will produce zero, 11.
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Postby Dubravsky » Sat Sep 10, 2011 1:36 pm

4.) Good! You may find interesting to note that what you recognised by summing odd numbers are actually simply square numbers, e.g. 16 = 4*4 = 1+3+5+7, 25= 5*5 = 1+3+5+7+9 etc. You may amuse yourself by proving the general fact (you can understand it simply by drawing dots).

5.) 1, 112, 112123

First entry has 111 added to it, and that's all (obviously). Second entry has 11 added to it and then it is concatenated with its former self ('112'+'123'='112123'). To get fourth entry you just add 1. Did I understand you correctly? Doesn't this sound like the a., b., c. paragraphs in my previous post? :)

I must admit though there are two good solutions here (I created this quickly), but you hit neither of them.

6.) Question your assumptions. Find a real rule instead of just "it's something descending and ...".

Tell me if you want more articulate hints.
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Postby tvelection » Sun Sep 11, 2011 3:11 pm

Dubravsky

You make good points about some of my answers/explanations. As for #6, the last sequence ---with positives and negatives, you are right. I made a deduction instead of defining a relation-rule (through deduction). Interestingly enough I thought I had a rule but it lost consistency (that is: the positives are produced by multiplying the ones and tens 49, (4x9) = 36, (3x6) = 18 but at (1x8) = 8, given 8 there is no subtraction to zero). However I do notice that 18 (-7) will get to 11, but will have to find a relation to make that relevant or otherwise disregard it as a coincidence --which I think it may be, . . . so far.

As for #5 (A+111, AB+11, ABC+1). Does seem contrived, especially since the the first +111 is (inconsistently) not "tacked" on to the first entry yielding 1112. Also there is no further continuation possible after +1. But that may be true of #6 at 0. Should I disregard any assumption, in number series problems, that the series may or may not possibly continue after the final entry? Or rather is the last entry sometimes a terminus of the series?

Anyway, no hints yet, let me look closer and see if I find more. Our exchange leads to an interesting circumstance of testing and accuracy; that is, if a test was only based on giving the right answer it might be less accurate (made with some deductions instead of figured rules for instance). Whereas if the test-giver required the test-taker to explain his or her answers it could become apparent that they had the right answer for the wrong (or incomplete) reasons.
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Postby tvelection » Mon Sep 12, 2011 3:31 pm

Dubravsky,

You've stumped me on #5 and I cannot find the rule for the "descending positives" to get to 11. Such tests are usually timed and I've looked now for 20 minutes, it may come in 20 more minutes, days, or not at all. I'm not picking up the gist of the rules of combination in those two instances though.

In working with #5) 1, 112, 112123, ?

I see "suspicious patterns" involving 1s but cannot connect them with the increase of total digits in the progression from 1 to 3 to 6 (which is sometimes irrelevant too). At one point I thought maybe 1, 1+1=2, 1+2=3, 1+3=4 yielding 112123134, but I'm not satisfied that it is a "strong" answer.

As for the rule I haven't found in #6) 49, -5, 36, -3, 18, ?, ?, 0

If the rule of subtracting ones from tens (4-9) to get the negatives holds then I can't get to 11 after (1-8 =, -7) , 11 without creating new rule that would establish (18-7) in that instance and given the former rule 11 (or any double) must come immediately before zero. Also interesting is that 4*9 is 36, and 3*6 is 18 which would yield 1-8 = -7 and then 1*8 =8, with no explanation of how 8 arrives at zero.

So good job to you too, I can't find the proper patterns there.
Anyway rather than give a hint maybe let it stand for a while so another member (or if I see it) can give the last two a better shot. I'll repost your series here.

Dubravsky wrote
More sequences:

4.) 1, 000X00000, 9, X0000X000, ?, ?

5.) 1, 112, 112123, ?

6.) 49, -5, 36, -3, 18, ?, ?, 0


Anyone else care to try #5 and #6 with a full explanation of the consistent rules at work?
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Postby Dubravsky » Tue Sep 13, 2011 11:26 am

I rather give the hints now. If someone else wants to chime in, just post a beep, comment, an own problem or anything so that we know someone's watching.

Hint for 6.) : Can you describe what happens when you subtract 9 from 4 to get -5 without reffering to tens and units?

Hint for 5.) :

1, 112, 112123, ?, ... : 1, 12, 123, 1234 ... = 1, 12, 123, 1234, ... : 1, 2, 3, 4, ...
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Postby tvelection » Wed Sep 14, 2011 12:54 pm

As for #5 . . . Each number shows up once, then, twice and when a number has appeared twice the next higher number is added to the string. And the former numbers' repetitions all increase by one. So there is 1 "one" then 2 "ones" (with 2 then added). Then there are 3 "ones" and 2 "twos" (3 is then added onto the end). This process yields:

1,112, 112123, 1121231234, and further 112123123412345

And in #6 you asked, "what happens when 9 is subtracted from 4," not sure what you are getting at because the question is seemingly so simple that it does not lead me to anything but some elementary answers. When you subtract 9 from 4: You move 9 intervals leftward on the number line. First you move 4 to 0 then 5 more to negative five. Or one may say when you subtract 9 from 4 you add -9 to 4. You may expect a different answer though. Everything seems consistent in that though 18 would be followed by -7, the last number before zero is what I'm trying to produce with rules to get something (if 11, or not) with complete consistency leading to 0.

No more hints on that just yet. I'm unable to pay for internet access at this point in my budget; so I post from the library or free Wi-Fi parking lots when the library is closed. Sometimes I can respond instantly sometimes it takes a day or two.
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Postby hyper » Wed Sep 14, 2011 5:38 pm

What is the meaning of the logical nummber sequence that you are demonstrarting ?
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Postby Dubravsky » Thu Sep 15, 2011 3:19 am

tvelection wrote:As for #5 . . . Each number shows up once, then, twice and when a number has appeared twice the next higher number is added to the string. And the former numbers' repetitions all increase by one. So there is 1 "one" then 2 "ones" (with 2 then added). Then there are 3 "ones" and 2 "twos" (3 is then added onto the end). This process yields:

1,112, 112123, 1121231234, and further 112123123412345


Good, but is that a complicated way to describe what's happening! It's sort of nested counting, isn't it? More precisely: Consider a function CountUpTo(n) that returns a list {1,2,...,n} when input positive integer n. Furthermore the function threads itself over lists, so that you can input a list such as {3,7} to get CountUpTo({3,7}) = { {1,2,3}, {1,2,3,4,5,6,7} }. Now the n-th term in our sequence is CountUpTo(CountUpTo(n)) (only the digits, concatenated). Example: CountUpTo(4) = {1,2,3,4}
CountUpTo(CountUpTo(4)) = CountUpTo({1,2,3,4}) = {CountUpTo(1), CountUpTo{2}, CountUpTo{3}, CountUpTo(4)} = { {1}, {1,2}, {1,2,3}, {1,2,3,4} } ----> 1121231234

Can you find the other solution (the one I didn't intend) here? It is closer to your original way of thinking on this item, but it's a consistent rule you see at work two times and yields a different answer than the one above.

hyper wrote: What is the meaning of the logical nummber sequence that you are demonstrarting ?

Hello hyper, which one do you mean? The one described in this post?
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Postby tvelection » Sat Sep 17, 2011 11:28 am

I'd like to say I've figured out #6 and the #5 alternate by now but I haven't, even with the hints. Some number sequences involve math manipulations others involve algorithms, procedures dealing with repetition, order, or sequencing rules. Having said that, I haven't found the alternate for #5, although the low numbers seem suspicious (especially the ones and the number 11). What throws me, and is a point of interest, is the first jump from 1 to 112.

As for your #6, "(49, -5, 36, -3, 18, ?, ?, 0)" and to your question, when one subtracts two numbers they find the difference between the two. Even if the numbers are reversed the absolute value is the same, e.g. 9-4 and 4-9. I've come to accept that the first question mark (from left to right) is -7, as it seems totally consistent with the way the other negatives are produced 18 (1-8 ). As for the positives, they would seem to be a multiplication of the previous positives (4*9) = 36 . . . and so (1*8 ) = 8, both together would yield . . . 18, -7, 8, 0. Of course, then I'm stuck explaining how 8 goes to 0 based on the rules established; and even though (8*0) = 0 there's no necessary justification for me to do that. I could use my previous tens/ones law and say (0*8 ) but you mentioned I was on the wrong track using tens and units.

Similar to what you've said in your opening to this thread, the simplicity has baffled me. These problems are interesting because when one finally sees or looks at the answer they are stunned at how easy it seems (after the fact we like to believe we 'could have'). If complexity can fly over one's head, then simplicity can slide under one's feet. It's easy for analytical thinkers to make simple concepts much more complicated by overlooking the obvious.

If you want you can give the answers, if not, I understand and will wait for others to try.
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