perplexus.info :: Numbers : Subsquares

Subsquares (Posted on 2008-05-21)

The numbers from 1 to 100 are arranged, snaking back and forth from the bottom, in a 10x10 grid:

100 99 98 97 96 95 94 93 92 91

 81 82 83 84 85 86 87 88 89 90

 80 79 78 77 76 75 74 73 72 71

 61 62 63 64 65 66 67 68 69 70

 60 59 58 57 56 55 54 53 52 51
+--------------+
|41 42 43 44 45|46 47 48 49 50
|              |
|40 39 38 37 36|35 34 33 32 31
|              |         
|21 22 23 24 25|26 27 28 29 30
|              |         
|20 19 18 17 16|15 14 13 12 11
+--+           |
| 1| 2  3  4  5| 6  7  8  9 10
+--+-----------+

Marked off are two subsquares that, if you add up the numbers within them, the total is the square of one of the numbers within. The larger square (5x5) totals 625, the square of 25, which is found within that marked square. The other one shown is the trivial "1", a 1x1 square, the single number in which of course is its own square.

Find another subsquare where the sum of its contained numbers is equal to the square of one of those numbers.

Submitted by Charlie

Rating: 5.0000 (1 votes)

Solution: (Hide)

100 99 98 97 96 95 94 93 92 91 81 82 83 84 85 86 87 88 89 90 80 79 78 77 76 75 74 73 72 71 61 62 63 64 65 66 67 68 69 70 60 59 58 57 56 55 54 53 52 51 41 42 43 44 45 46 47 48 49 50 +--------+ 40 39 38 37 36|35 34 33|32 31 | | 21 22 23 24 25|26 27 28|29 30 | | 20 19 18 17 16|15 14 13|12 11 +--------+ 1 2 3 4 5 6 7 8 9 10

The marked square adds up to 225, the square of 15, a number within it.

CLS FOR stRow = 1 TO 9 STEP 2 v = 100 - stRow * 10 + 10 FOR col = 1 TO 10 b(stRow, col) = v v = v - 1 NEXT FOR col = 10 TO 1 STEP -1 b(stRow + 1, col) = v v = v - 1 NEXT NEXT FOR r = 1 TO 10 FOR c = 1 TO 10 PRINT USING "###"; b(r, c); NEXT PRINT : PRINT NEXT FOR size = 1 TO 10 FOR top = 1 TO 10 - size + 1 FOR lft = 1 TO 10 - size + 1 tot = 0 FOR i = top TO top + size - 1 FOR j = lft TO lft + size - 1 tot = tot + b(i, j) NEXT NEXT sr = INT(SQR(tot) + .5) IF sr * sr = tot THEN FOR i = top TO top + size - 1 FOR j = lft TO lft + size - 1 IF b(i, j) = sr THEN PRINT top, lft, size PRINT sr, tot END IF NEXT NEXT END IF NEXT NEXT NEXT

Based on Enigma No. 1490, "Board with squares?", by Susan Denham, New Scientist, 19 April 2008

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

Subject	Author	Date
No more solutions, I think	pcbouhid	2008-05-23 14:28:04
four by four and on	Paul	2008-05-21 21:20:42
two by two	Paul	2008-05-21 20:50:41
Little square	charles Bohnsack	2008-05-21 18:16:56
re: Who's on First?	Charlie	2008-05-21 12:30:15
solution(s)	Dej Mar	2008-05-21 12:24:07
Who's on First?	ed bottemiller	2008-05-21 11:57:18
One of them (3 x3)	pcbouhid	2008-05-21 11:11:07
One Missing - Roman.	brianjn	2008-05-21 11:05:49

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