perplexus.info :: Numbers : January Math Challenge in 3 Parts

January Math Challenge in 3 Parts (Posted on 2009-01-08)

Not often does a novel contain math problems. All Souls, by Christine Schutt, does. Here's a section of the chapter called "Numbers":

Siddons
The January Math Challenge had three parts:

Part I: Express as many of the perfect squares less than 1,000 as the sums of two or more consecutive integers as possible. Example: 9 = 4 + 5
Part II: The sequence 2, 3, 5, 6, 7, 8, 10, 11 consists of all positive integers that are not perfect squares. What is the 500th term of the sequence?
Part III: Find the largest positive integer such that each pair of consecutive digits forms a perfect square. Example: The number 364 is made up of the perfect squares 36 and 64.

Saperstein and Song both said it was easy.
"Then why don't you answer it and get a bag of M&M's?" Alex asked.
"Because we helped make it up."

Assume that in Part I, "consecutive integers" means "consecutive positive integers". There should also be an ellipsis ("...") after the 11 in Part II, to indicate the series continues indefinitely.

See how well you do on the "January Math Challenge".

Submitted by Charlie

Rating: 3.0000 (3 votes)

Solution: (Hide)

Part I:

4 + 5 = 9 78 + ... + 82 = 400 2 + ... + 4 = 9 4 + ... + 28 = 400 12 + 13 = 25 220 + 221 = 441 3 + ... + 7 = 25 146 + ... + 148 = 441 71 + ... + 76 = 441 11 + ... + 13 = 36 60 + ... + 66 = 441 1 + ... + 8 = 36 45 + ... + 53 = 441 25 + ... + 38 = 441 24 + 25 = 49 16 + ... + 33 = 441 4 + ... + 10 = 49 11 + ... + 31 = 441 40 + 41 = 81 57 + ... + 64 = 484 26 + ... + 28 = 81 39 + ... + 49 = 484 11 + ... + 16 = 81 5 + ... + 13 = 81 264 + 265 = 529 12 + ... + 34 = 529 18 + ... + 22 = 100 9 + ... + 16 = 100 191 + ... + 193 = 576 60 + ... + 68 = 576 60 + 61 = 121 6 + ... + 16 = 121 312 + 313 = 625 123 + ... + 127 = 625 47 + ... + 49 = 144 58 + ... + 67 = 625 12 + ... + 20 = 144 13 + ... + 37 = 625 84 + 85 = 169 81 + ... + 88 = 676 7 + ... + 19 = 169 46 + ... + 58 = 676 25 + ... + 31 = 196 364 + 365 = 729 21 + ... + 28 = 196 242 + ... + 244 = 729 119 + ... + 124 = 729 112 + 113 = 225 77 + ... + 85 = 729 74 + ... + 76 = 225 32 + ... + 49 = 729 43 + ... + 47 = 225 14 + ... + 40 = 729 35 + ... + 40 = 225 21 + ... + 29 = 225 109 + ... + 115 = 784 18 + ... + 27 = 225 9 + ... + 40 = 784 8 + ... + 22 = 225 4 + ... + 21 = 225 420 + 421 = 841 15 + ... + 43 = 841 144 + 145 = 289 9 + ... + 25 = 289 299 + ... + 301 = 900 178 + ... + 182 = 900 107 + ... + 109 = 324 109 + ... + 116 = 900 37 + ... + 44 = 324 96 + ... + 104 = 900 32 + ... + 40 = 324 53 + ... + 67 = 900 2 + ... + 25 = 324 26 + ... + 49 = 900 24 + ... + 48 = 900 180 + 181 = 361 3 + ... + 42 = 900 10 + ... + 28 = 361 480 + 481 = 961 16 + ... + 46 = 961 for the above: OPEN "perfsq2.txt" FOR OUTPUT AS #2 FOR sr = 2 TO 31 sq = sr * sr n = 2 DO st = 0 mid2 = 2 * sq / n IF mid2 = INT(mid2) THEN mid = mid2 / 2 good = 0 IF mid = INT(mid) AND n MOD 2 = 1 THEN st = mid - (n - 1) / 2: good = 1 ELSEIF mid <> INT(mid) AND n MOD 2 = 0 THEN st = mid + .5 - n / 2: good = 1 END IF IF st > 0 AND good = 1 THEN IF psq < sq THEN PRINT #2, PRINT #2, st; IF n > 2 THEN PRINT #2, "+ ... +"; : ELSE PRINT #2, "+"; PRINT #2, st + n - 1; "="; sq ' FOR i = st TO st + n - 1: PRINT #2, i; : NEXT ' PRINT #2, " == "; sq: PRINT #2, psq = sq END IF END IF n = n + 1 LOOP UNTIL st < 0 OR n > sq NEXT

Part II:
If the perfect squares were not excluded, the 500th number would be 500. But there are [sqrt(500)] = 22 perfect squares under 500, where the square brackets indicate the floor function. So 522 is the answer unless another perfect square intervenes between 500 and 522, which it does not. So the final answer is 522.
Part III:
81649

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Answer	K Sengupta	2023-10-20 06:55:45
Puzzle Thoughts	K Sengupta	2023-06-25 02:24:57
Part 1 (non compute solution)	Gamer	2009-01-08 16:20:46
Part II (Non-computer solution)	rod hines	2009-01-08 14:17:48
Part II (computer solution)	Daniel	2009-01-08 13:43:05
Part I (computer solution)	Daniel	2009-01-08 13:39:19
Part III (spoiler)	Steve Herman	2009-01-08 13:11:28

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