Wikipedia:Reference desk/Archives/Mathematics/2023 December 5

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December 5

Playing with Pascal's triangle

One of the most interesting questions about Pascal's triangle is:

Keep the numbers in each row to a minimum as long as the following criteria are met:

1. Terms of each row are strictly increasing as we go towards the middle.
2. For every a,b,c where a and b are 2 consecutive terms of a row in the triangle and c is the term immediately below, there are no such a,b,c where exactly 2 of a,b,c have a common prime factor.
3. No number should be divisible by k until the k+1th row of the triangle if k is a prime power.

Let me try:

First 3 rows

The first row is 1. The second row is 1,1. This is easy. The third row?? It can't be 1,1,1 because the middle needs to be larger than the sides; check 1,2,1 and it is correct!

Fourth row

The fourth row is 1,3,3,1. No problem yet!

Fifth row

The fifth row cannot be 1,4,5,4,1 because 3,3,5 fails to meet the second criterion (3 and 3 are both multiples of 3 but 5 is not.) This leads us to 1,4,6,4,1; this is correct!

Sixth row

Now let's consider the sixth row. The 1,5 at the beginning and 5,1 at the end are correct indeed, but how about those in the middle?? 4 and 6 have a GCF of 2, so the number must be a multiple of 2. Check 8. 4 and 8 are both multiples of 4 but 6 is not, so it's wrong. Check 10 now. 1,5,10,10,5,1 is correct!

Seventh row

1,6,A,B,A,6,1; what are A and B?? Now, A must be a multiple of 5 because the GCF of 5 and 10 is 5. Check 15; this is correct! B must be a multiple of 10; check 20; it is correct! So the seventh row (1,6,15,20,15,6,1) we got!

Eighth row

For finding A and B in 1,7,A,B,B,A,7,1 we first have to note that A must be a multiple of 3 (the GCF of 6 and 15.) 18?? Wrong, because 6 and 18 have a common factor of 6. 21?? Right. Now, for finding B here, this is the first time the third criterion is necessary. 15 and 20 have a GCF of 5, so we can try 25. It meets the first 2 criteria, but not the third. Why?? 25 is a multiple of 25 (a prime power) and we're not allowed to have any multiples of 25 until the 26th row. Not 30 because 15 and 30 have a common factor of 10. Is 35 right?? Yes! 1,7,21,35,35,21,7,1

Ninth row

1,8,A,B,C,B,A,8,1. A must be a multiple of 7 and 28 is correct! B must be a multiple of 7; 42 is wrong because 21 and 42 have a GCF greater than 7; 49 is wrong because we can't have any multiples of 49 until the 50th row; 56 is right! C must be a multiple of 35, and 70 is correct! 1,8,28,56,70,56,28,8,1

Tenth row

1,9,A,B,C,C,B,A,9,1. A must be a multiple of 4; 32 is wrong because we can't have any multiples of 32 until the 33rd row. 36 is correct! B must be a multiple of 28 and 84 is correct! C must be a multiple of 14; 84 is wrong because 84 and 56 have a GCF greater than 14 (and also it violates the rule that terms strictly increase as we get closer to the middle of the row); 98 is wrong because we can't have any multiples of 49 until the 50th row; 112 is wrong because we can't have any multiples of 16 until the 17th row; 126 is right! 1,9,36,84,126,126,84,36,9,1

Eleventh row

1,10,A,B,C,D,C,B,A,10,1. A must be a multiple of 9 and 45 is right! B must be a multiple of 12. 96 is wrong because we can't have any multiples of 32 until the 33rd row. 108 is wrong because we can't have any multiples of 27 until the 28th row. 120 is correct! C must be a multiple of 42. 168 is wrong because 84 and 168 have a GCF > 42. 210 is correct! D is clearly 252. So we got the eleventh row correctly; 1,10,45,120,210,252,210,120,45,10,1

Twelfth row

1,11,A,B,C,D,D,C,B,A,11,1. A must be a multiple of 5. 50 is wrong because we can't have any multiples of 25 until the 26th row. 55 is correct! B must be a multiple of 15. 135 is wrong because we can't have any multiples of 27 until the 28th row. 150 is wrong because we can't have any multiples of 25 until the 26th row. 165 is correct! C must be a multiple of 30. 240 is wrong because 120 and 240 have a GCF > 30. 270 is wrong because we can't have any multiples of 27 until the 28th row. 300 is wrong because 120 and 300 have a GCF > 30. 330 is correct! D must be a multiple of 42. 336 is wrong because 252 and 336 have a GCF > 42. 378 is wrong because we can't have any multiples of 27 until the 28th row. 420 is wrong because 210 and 420 have a GCF > 42. 462 is correct! The row is 1,11,55,165,330,462,462,330,165,55,11,1.

Thirteenth row

1,12,A,B,C,D,E,D,C,B,A,12,1. A must be a multiple of 11 and 66 is correct! B must be a multiple of 55 and 220 is correct! C must be a multiple of 165 and 495 is correct! D must be a multiple of 66. 528 is wrong because we can't have any multiples of 16 until the 17th row. 594 is wrong because we can't have any multiples of 27 until the 28th row. 660 is wrong because 330 and 660 have a GCF > 66. 726 is wrong because we can't have any multiples of 121 until the 122nd row. 792 is correct! E must be a multiple of 462, and 924 is correct! We have the row 1,12,66,220,495,792,924,792,495,220,66,12,1.

Fourteenth row

1,13,A,B,C,D,E,E,D,C,B,A,13,1. A must be a multiple of 6; 72 is wrong because 12 and 72 have a GCF > 6; 78 is correct! B must be a multiple of 22. 242 is wrong because we can't have any multiples of 121 until the 122nd row. 264 is wrong because 66 and 264 have a GCF > 22. 286 is correct! C must be a multiple of 55. 550 is wrong because 220 and 550 have a GCF > 55. 605 is wrong because we can't have any multiples of 121 until the 122nd row. 660 is wrong because 220 and 660 have a GCF > 55. 715 is correct! D must be a multiple of 99. 891 is wrong because we can't have any multiples of 81 until the 82nd row. 990 is wrong because 495 and 990 have a GCF > 99. 1089 is wrong because we can't have any multiples of 121 until the 122nd row. 1188 is wrong because 792 and 1188 have a GCF > 99. 1287 is correct! E must be a multiple of 132. 1320 is wrong because 792 and 1320 have a GCF > 132. 1452 is wrong because we can't have any multiples of 121 until the 122nd row. 1584 is wrong because 792 and 1584 have a GCF > 132. 1716 is correct! We now have the fourteenth row of the triangle correct; it is 1,13,78,286,715,1287,1716,1716,1287,715,286,78,13,1.

Fifteenth row

1,14,A,B,C,D,E,F,E,D,C,B,A,14,1. A must be a multiple of 13, and 91 is correct! B must be a multiple of 26. 312 is wrong because 78 and 312 have a GCF > 26. 338 is wrong because we can't have any multiples of 169 until the 170th row. 364 is correct! C must be a multiple of 143. 858 is wrong because 286 and 858 have a GCF > 143. 1001 is correct! D must be a multiple of 143. 1430 is wrong because 715 and 1430 have a GCF > 143. 1573 is wrong because we can't have any multiples of 121 until the 122nd row. 1716 is wrong because 1287 and 1716 have a GCF > 143. 1859 is wrong because we can't have any multiples of 169 until the 170th row. 2002 is correct! E must be a multiple of 429. 2145 meets all the above criterion and is wrong. The correct number is 3003. This is finally the answer to one of the most interesting questions about Pascal's triangle. It is the first point in Pascal's triangle where there's a value smaller than the true value that meets 3 notable criteria, but is wrong; specifically 2145 as opposed to 3003. (As for why this is interesting, please note that I just find it interesting to explore Pascal's triangle and look for interesting questions to answer. (Please make sure you know all 3 criteria. With just the first 2 criteria, the answer would be 25 as opposed to 35 in the eighth row. If possible, please feel free to check the above for any corrections that need to be made.) Any more interesting question about Pascal's triangle?? Georgia guy (talk) 21:26, 5 December 2023 (UTC)

Your solution for the first three rows is the least one. I do not see how the criteria preclude the following from also being a solution:

     1

   1   1

1   4   1

 --Lambiam 18:42, 6 December 2023 (UTC)

Lambiam, the rule is that I try to keep the values as small as possible. Because 2 < 4, 2 needs to be checked first. Georgia guy (talk) 18:50, 6 December 2023 (UTC)

Sorry, I did not read the requirements carefully enough.  --Lambiam 07:46, 7 December 2023 (UTC)

I don't see why given those rules the fifth row couldn't be 1 2 3 2 1. That seems to follow all the rules. --92.40.47.68 (talk) 20:16, 6 December 2023 (UTC)

It's a fact that takes hardly any skill that the terms adjacent to the 1's are always simply the sequence of natural numbers; that is, 2,3,4,5... If the fifth row were 1,2,3,2,1 the diagonal would go 1,2,3,2.... Georgia guy (talk) 21:05, 6 December 2023 (UTC)

Can you say which specific criterion is violated?  --Lambiam 07:46, 7 December 2023 (UTC)

Lambiam, the criterion is that the numbers strictly increase for each row. (Barring the 1's at the sides, of course!) (I thought this one was obvious initially; I wasn't thinking of the "1,2,3,2,1" response.) Georgia guy (talk) 11:17, 7 December 2023 (UTC)

One stated criterion is that the terms of a row strictly increase as we go towards the middle, which they do in "1 2 3 2 1". Is this criterion meant to impose a condition on terms in a given row in relation to terms in earlier rows as well?. Using mathematical notation for the criteria may help to avoid unintended interpretations.  --Lambiam 12:15, 7 December 2023 (UTC)

Lambiam, yes they do. Also very obvious is that each term adjacent to a 1 in each row must be just one less than the row's number; that is, the fifth row starts with 1,4... and ends with ...4,1. Georgia guy (talk) 12:23, 7 December 2023 (UTC)

Based on row 6 and the problem being a GCD of 4, I presume that your second rule should be about common prime power factors, right? GalacticShoe (talk) 15:37, 7 December 2023 (UTC)

Yes, it's about prime power factors. 1-4-5-4-1 isn't valid for the fifth row because 3 and 3 (on the fourth row) have GCF of 3 and the number in the middle of the fifth row has to be a multiple of 3. 1-5-8-8-5-1 isn't valid for the sixth row because 4 and 6 (on the fifth row) have a GCF of 2 and 8 has a larger factor than 2 in common with one of those (specifically 4.) Georgia guy (talk) 15:44, 7 December 2023 (UTC)

In that case, we can formalize the three rules by defining, over the entries of the triangle, a function $Q(r,c)$ for rows $r\geq 1$ and "columns" $1\leq c\leq r$, which takes the minimum integer given these stipulations:

1. $Q(r,1)=1$, $Q(r,c)>Q(r,c-1),Q(r-1,c-1),Q(r-1,c)$ for $1<c\leq \lceil {\frac {r}{2}}\rceil$, and $Q(r,c)=Q(r,(r+1)-c)$ for $c>\lceil {\frac {r}{2}}\rceil$

2. Letting $\nu _{p}(n)$ be the highest power of $p$ dividing $n$ (i.e., its p-adic valuation), $\nu _{p}(Q(r-1,c-1))\neq \nu _{p}(Q(r-1,c))\Rightarrow \nu _{p}(Q(r,c))=\min(\nu _{p}(Q(r-1,c-1)),\nu _{p}(Q(r-1,c)))$

3. Otherwise, $\nu _{p}(Q(r-1,c-1))=\nu _{p}(Q(r-1,c))\Rightarrow \nu _{p}(Q(r-1,c-1)),\nu _{p}(Q(r-1,c))\leq \nu _{p}(Q(r,c))\leq \log _{p}(r-1)$

GalacticShoe (talk) 15:55, 7 December 2023 (UTC)

GalacticShoe, did I get the point where this triangle deviates from Pascal's triangle right?? Georgia guy (talk) 17:08, 7 December 2023 (UTC)

I wrote up a quick computer program and your values are all correct, and indeed this is the point where the triangle deviates from Pascal's triangle. The last value in the row, $3432$, actually once again matches Pascal's triangle though. But as if it is unsure of what to do with this one broken value, the triangle's next row breaks down. All of the first few values match Pascal's triangle (they are $1,15,105,455,1365,3003$), until where you would normally expect to see $2002+3003=5005$. At this point, the two numbers above are instead $2002$ and $2145$. Since $2002=2*7*11*13$ and $2145=3*5*11*13$, the next number must be a multiple of $11*13=143$, but not a multiple of $2,3,5,7$. In our number, we cannot use $17$ or higher since we are on row $16$, and we also cannot use any power of $11$ or $13$ greater than what we already have, so the triangle breaks down. GalacticShoe (talk) 17:47, 7 December 2023 (UTC)

GalacticShoe, the sum, 4147, is 11 times 13 times 29. The really interesting question here is: can you add a fourth stipulation that would make the triangle not deviate until a later point?? Georgia guy (talk) 18:00, 7 December 2023 (UTC)

Although there are certainly arbitrary conditions one can impose to allow the triangle to not deviate until a further point, there are none that are simple enough that particularly stand out to me, short of the additivity condition that literally defines Pascal's triangle. GalacticShoe (talk) 18:17, 7 December 2023 (UTC)