# Sequence puzzle

This is a sequence puzzle I found online which I'm stuck on:

452801, 773924, 102410, 471056, ?

I figured out how to get from the first number to the second (+3, +2, +1, +1, +2, +3) for the digits, but this rule doesn't work for the rest. Any ideas?

• 050321 might be next number in sequence – CR241 Oct 3 '18 at 23:54
• I checked, it's not @CR241 – Duck Oct 25 '18 at 23:35
• I haven't had the time yet, but seems like it might be a determined value being added each time, but the sum is capped as would a variable in software be (example, if you do addition with an integer 16-bit value and the answer is above the cap, it literally cuts off the highest value digits to keep within the integer 16-bit value range) – nine9 Mar 1 '19 at 8:28

I believe @Adam already found the correct solution, but I may have some new ideas concerning the pattern:

If we start with the number $$452801$$ and then apply the function (+3, +2, +1, +1, +2, +3) to the last 6 digits of each number, we obtain the sequence:
$$452801$$ > $$773924$$ > $$10941047$$ > $$1012621610$$ > $$1012942733$$

Replacing the 9 in the third number by a 2 yields:
$$452801$$ > $$773924$$ > $$10241047$$ > $$105621610$$ > $$105942733$$

Regrouping to 6-digit numbers we get:
$$452801$$ > $$773924$$ > $$102410$$ > $$471056$$ > $$216101$$ > $$059427$$ > $$33....$$

Which is exactly the sequence we're looking for. Unfortunately we needed to magically replace one digit by another in the process... This could of course just be a typo, but there might also me something else going on that I'm still missing. The overall pattern, however, is fitting too well for it to be a complete coincidence, I feel.

• 100% agree with the concatenation of numbers from the +3 +2 +1 +1 +2 +3, this definitely has to be apart of the final solution. That 9 is really odd for such a convenient setup that you found. I wouldn't be surprised if it was an error at this point (+1) – Adam Jul 15 '19 at 17:17
• Should it be 105621610 + 321123 = 105942733 for the last term? – simonzack Jul 16 '19 at 14:40
• @simonzack ah yes, you're right, I'll edit this – Levieux Jul 16 '19 at 14:42
• Thanks for this finding, reckon this was the intended answer! I'm guessing it's a typo as you said and marking it as accepted. Wasn't the best outcome to be a typo haha. – simonzack Jul 16 '19 at 14:44

The next number in the sequence is:

$$216101$$

I got it by creating an algorithm that tested positive integers until a result was generated indicating that the question was answered correctly. The algorithm submitted an answer one at a time and waited for a response before submitting the next answer to make sure that the website wasn't overwhelmed.

For the curious - it took about 5 hours.

Hey, @Mr Pie's strange observation is correct!

## An attempt to find a correlation between the numbers

$$452801 \Rightarrow 773924$$, $$[4+3,5+2,2+1,8+1,0+2,1+3]=[7,7,3,9,2,4]$$

$$773924 \Rightarrow 102410$$, $$[7+3,7+2,3+1,9+1,2+2,4+3]=[10,9,4,10,4,7]$$ $$[10,9+4+4+7,10]=[10,24,10]=[1,0,2,4,1,0]$$

$$102410 \Rightarrow 471056$$,$$[1+3,0+2,2+1,4+1,1+2,0+3]=[4,2,3,5,3,3]$$ $$[4,4+3,2+3+5,5,3+3]=[4,7,10,5,6]=[4,7,1,0,5,6]$$

$$471056 \Rightarrow 216101$$,$$[4+3,7+2,1+1,0+1,5+2,6+3]=[7,9,2,1,7,9]$$ $$[2,7+7+2,7+2+1,1]=[2,16,10,1]=[2,1,6,1,0,1]$$

Just wanted to get this out there because I feel very confident about the first two correlations although the next two are very flimsy. Maybe this edit will generate more interest in this question and someone will finally find the answer

• Hahah thank you! I thought about doing it myself but never did it. Thanks for doing it for me. I upvoted but haven't accepted your answer yet since like you said you haven't found a pattern yet. – simonzack May 3 '19 at 10:57
• Oh, what do you know! :P – Mr Pie May 5 '19 at 2:58

This is not exactly a pattern, but there is one odd thing these numbers have in common:

If you get each term and divide it by $$\sqrt{e^\pi}$$ then it is close to an integer.

Proof (correct to $$5$$ decimal places):

$$452801\div\sqrt{e^\pi}=94128.08005\approx 94128$$

$$773924\div\sqrt{e^\pi}=160822.99325\approx 160823$$

$$102410\div\sqrt{e^\pi}=21288.94741\approx 21289$$

$$471056\div\sqrt{e^\pi}=97922.92172\approx 97922$$

The answers are becoming less nearer to an integer the further down the sequence we continue, thus perhaps we have to add or subtract a constant to each term that correspondingly gets bigger, in order to make the answers become closer to an integer.

This might have to do with the overall pattern.

Hope this helps!

• Yea in Adam's answer we have $216101\div\sqrt{e^\pi}=44922.984328976$. – simonzack May 6 '19 at 6:21

One thing all 4 numbers have in common is this :

Rule: They are all the last 6 digits of the integer part of the quarter-square of some integer.

where

The quarter square of a number $$x$$ is defined as $$(\frac{x}{2})^2$$ or $$\frac{x^2}{4}$$. A list of quarter-squares upto 100,000 can be found here.

For example

The quarter-square of the number 216,371 is :

$$(\frac{216371}{2})^2=11,704,102,410.25$$

The last 6 digits of the integer part are 102,410.

Going up to 2 million,

which is a 7-digit number, and also a period for the above rule,

we can find a list of numbers that adhere to the above rule :

452801 : [69698, 492802, 507198, 930302, 1069698, 1492802, 1507198, 1930302]

773924 : [54764, 179764, 320236, 445236, 554764, 679764, 820236, 945236, 1054764, 1179764, 1320236, 1445236, 1554764, 1679764, 1820236, 1945236]

102410 : [216371, 627379, 1372621, 1783629]

471056 : [53832, 76065, 94815, 196168, 303832, 305185, 323935, 446168, 476065, 494815, 553832, 696168, 705185, 723935, 803832, 876065, 894815, 946168, 1053832, 1105185, 1123935, 1196168, 1276065, 1294815, 1303832, 1446168, 1505185, 1523935, 1553832, 1676065, 1694815, 1696168, 1803832, 1905185, 1923935, 1946168]

The above lists were calculated using

this code written in Swift.

But I still can't find a relationship between the numbers in those lists.

• Wow nice a lot of work I upvoted cause of that, still not sure about the answer. – simonzack Mar 30 '19 at 13:58
• How did you find that?! :o – Mr Pie Apr 8 '19 at 3:36
• Yes, would love to know how you knew about the last 6 digits of a quarter square... not exactly common mathematical knowledge – Parseltongue Jul 14 '19 at 21:42

I am not quite sure but was able to make some quite good relation

1. 452801=a , 773924=b , 102410=c , 471056=d
2. Now let us take b-a=321123=A , d-c=368646=B , b-c=671514=C , d-a=18255=D