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WHOA! COOL MATH CURIOUS MATHEMATICS FOR FUN AND JOY NOVEMBER 2012 PROMOTIONAL CORNER: Have you an event, a workshop, a website, some materials you would like to share with the world? Let me know! If the work is about deep and joyous and real mathematical doing I would be delighted to mention it here. The square numbers arise from arranging pebbles into square arrays: and the triangular numbers from making triangular arrays: *** Looking for high-end, but completely accessible, rich, relevant, and meaningful explorative experiences for your students? Would you like it all coupled with guidance for teachers too? Check out: MAKING MATHEMATICS: Mentored Research Projects for Young Mathematicians. http://www2.edc.org/makingmath/mathp roj.asp You and your students are in for a treat! This month’s essay simply asks: Are there more square numbers than triangular numbers? Are there more triangular numbers than square numbers? Or are the counts of each essentially the same? Given that there are infinitely many of each type of number is this question meaningless? Hmm. © James Tanton 2012 ASIDE: Some interplay between square and triangular numbers. The following diagram shows that doubling any triangular number and adding to it its matching square number gives another triangular number: For example double 21 , the sixth triangular number, plus the sixth square number gives: 2 × 21 + 36 = 78 , the twelfth triangular number. This picture FORMULAS: Let S N denote the N th square number and TN the N th triangular number. We have: SN = N 2 TN = 1 + 2 + 3 + ⋯ + N This second formula is not particularly helpful. But noting that two triangles placed together make an oblong allows us to see that TN is half of N ( N + 1) We have: TN = N ( N + 1) 2 . Puzzler: Show, both algebraically and pictorially, that the triangular numbers satisfy: TM + N = TM + TN + MN . shows that the sum of two consecutive triangular numbers gives a square. For instance, 3 + 6 = 9 , 6 + 10 = 16 and 10 + 15 = 25 . (Why don’t two triangles the same size make a square?) Puzzler: Can you draw pictures to explain the following curiosities? Pick any three consecutive triangular numbers, multiply the middle one by six and add to this product the remaining two. The result is square number. Show, both algebraically and pictorially, that the square numbers satisfy: S M + N = S M + S N + 2 MN . HIGHER FIGURATE NUMBERS: The triangular numbers are formed successively adding rows to the following diagram: For instance: 3 + 6 × 6 + 10 = 49 . Pick any triangular number and multiply it by eight. The result is always one less than a square number. For instance: 8 × 15 + 1 = 121 . www.jamestanton.com © James Tanton 2012 The “pentagonal numbers” are formed by adjoining three diagrams and successively adding rows: COUNTING FIGURATE NUMBERS: Clearly there are infinitely many square numbers and infinitely many triangular numbers, and so asking which there are “more of” can, at first, seem meaningless. However, if we think of “more” as a question of “density” then there might be a difference between the two worth exploring. For instance there are 31 square numbers and 44 triangular numbers among the numbers 1 through 1000. These counts are different. In fact, the following table suggests there is something interesting afoot. (The number 1.414 is suspiciously like the square root of two!) And so on! Let’s explore “more of” in terms of density. The square numbers are formed by adjoining two of these diagrams and successively adding rows: Find a formula for the N th pentagonal number PN . Find a formula for the N th hexagonal number H N . Find a formula for the N th septagonal number GN . Show that: PM + N = PM + PN + 3MN H M + N = H M + H N + 4 MN GM + N = GM + GN + 5MN Question One: If N is a large number, how many numbers among 1, 2,3,..., N are square? The square numbers in this list are the numbers 1, 4,9,.., k 2 with k the largest integer satisfying k 2 ≤ N . That is, k is the largest integer ≤ N . That is, k is the value N rounded down to an integer. Mathematicians denote this N (and call it the floor of N ). We have: There are N square numbers among 1, 2,3,..., N . www.jamestanton.com © James Tanton 2012 Question Two: How many numbers among 1, 2,3,..., N are triangular? The triangular numbers we seek are k ( k + 1) 1,3, 6,..., with k as large as 2 1 possible satisfying k ( k + 1) ≤ N . So 2 we need to find the largest value k satisfying k 2 + k ≤ 2 N . Completing the square, we need the largest value of k with: 1 1 k 2 + k + ≤ 2N + 4 4 2 1 1 k + ≤ 2N + 2 4 1 1 − 4 2 This shows: There are 1 1 2 N + − triangular numbers 4 2 among 1, 2,3,… , N . k ≤ 2N + If N is indeed truly large, adding a quarter before taking a square root, will have little effect (the square roots of 2000000 and 2000000.25 are very close) and subtracting a half from a large answer will offer negligible change. 1 1 Thus 2 N + − is very close to the 4 2 number 2N , which looks about 2 as big as the formula N . We see that the count of triangular numbers is indeed larger than the count of square numbers by a factor close to 2 (and the error of this claim likely becomes negligible for larger and larger values of N ). In this sense we can say: There are 2 more triangular numbers than square numbers! Very technical aside: To make this claim iron-clad we need to establish: 1 1 2N + − 4 2 = 2. lim N →∞ N This can be done by noting that for any value x we have: x − 1 ≤ x ≤ x . Thus 1 3 2 N + 1 − 1 − 4 2 4 2≤ ≤ N N 2N + 1 1 − 4 2 N −1 2N + That is: 1 1 2N + − 4 2 1 3 − ≤ ≤ 2+ 4N 2 N N 1 1 − 4N 2 N 1 1− N 2+ As N → ∞ , the left side approaches 2 + 0 − 0 = 2 and the right side 2+0 −0 = 2 . The 1− 0 quantity in between, the ratio we seek, thus approaches the value 2 as well. (Ahhh. The “squeeze theorem” from calculus!) approaches RESEARCH CORNER 1: How many more triangular numbers are there than pentagonal numbers? Than hexagonal numbers? Than septagonals? EXTRA: THE SQUANGULARS! Here are the square and triangular numbers: Notice that 1 and 36 belong to both lists – they are both square and triangular. I call these numbers squangular. The next squangular number is 1225, and then comes 41616. In fact there are infinitely many squangular numbers and the N th squangular number is given by the crazy formula: www.jamestanton.com © James Tanton 2012 ( ) ( 3+ 8 N − 3− 8 2 8 ) N 2 [See www.jamestanton.com/?p=519 for an accessible derivation of this!] RESEARCH CORNER 2: Are there any pent-squangular numbers (apart from 1)? Is there a non-trivial number that is simultaneously square, triangular and pentagonal? [I personally do not know the answer to this.] numbers halfway between being nonsquare and non-triangular - whatever that means! Putting N = 1, N = 2, N = 3,... into N+ 1.5 N gives the sequence of “non-halfway-square/triangulars” 2, 4,5, 6,8,9,10,11,13,14,15,16,17,19, 20,... So the “actual halfway square/ triangulars” are the numbers not listed here, namely: 1,3, 7,12,18, 25, 33, 42, 52, 63,... SOMETHING BIZARRE! Here are the non-square numbers, everything but the squares: [This truly is absurd!] RESEARCH CORNER 3: It looks like the sequence 2,3, 5, 6, 7,8,10,11,12,13,14,15,17,... . 1,3, 7,12,18, 25, 33, 42, 52, 63,... The N th number in the list is given by the formula N + N . Here the angled brackets mean round up or down to the nearest integer. Challenge: Verify this formula for the non-squares. (Hint: We know that there are N squares among the numbers 1 through N , so the N th non-square is the number N “bumped up” N places. But we need careful and check if this bumping goes beyond yet another square number.) Here are the non-triangular numbers: 2, 4, 5, 7,8, 9,11,12,13,14,16,17,.... The N th one in the list is given by the formula N + 2N . Challenge: Verify this! has the property that every counting number either appears in the sequence or as the difference of two consecutive terms of the sequence. (The sequence of differences is back to 2, 4, 5, 6,8, 9,10,... by the way!) Is this actually the case? RESEARCH CORNER 4: Is there any meaningful way to actually interpret the sequence of numbers 1,3, 7,12,18, 25, 33, 42, 52, 63,... as halfway, in some manner, between the sequence of triangular numbers and the sequence of square numbers? [I haven’t seen any reason for it myself. If you do, let me know!] RESEARCH CORNER 5: Dare I ask about non-pentagonals, or sequences five-twelfths of the way between the squares and the septagonals? (Oh my!) This suggests that the formula N+ 1.5 N should correspond to © 2012 James Tanton [email protected] www.jamestanton.com