# An Enchanting Property Of Numbers Greater Than 60 And Ending In 0 Or 5

## An Enchanting Property Of Numbers Greater Than 60 And Ending In 0 Or 5

Background:  This is about a recently received email video, reference unavailable, in which a  TV program guest asks the host to tell him a number between 65 and 300 and which ends in 0 or 5. As the host responds with the number 265, the guest quickly, seemingly in no particular order, writes twenty five numbers on a 5 x 5 chart (Table). The guest then asks the host to add the numbers along any row, or a column, or a diagonal on the chart, which the host does, finding, much to the amazement of audience and himself, that numbers on the chart in any row, or column, or diagonal add to the original number (265).

Replication of the above, in three Steps (I), (II) and (III):

(I) The 5 x 5 chart (Table) and its elements:

The following represents a 5 x 5 Table (chart) with 25 boxes or locations (T11, T21, .., T55), in which the label T for any location / box has the first suffix (1, 2, 3, 4 or 5, after T) corresponding to the row number (R1, R2, R3, R4 or R5), and the second suffix (1, 2, 3, 4 or 5, after the row number) corresponding to the column number (C1, C2, C3, C4 or C5).

....... (C1) (C2) (C3) (C4) (C5)

(R5)   T51, T52, T53, T54, T55
(R4)   T41, T42, T43, T44, T45
(R3)   T31, T32, T33, T34, T35
(R2)   T21, T22, T23, T24, T25
(R1)   T11, T12, T13, T14, T15

(II) Determination of the lowest (or starting) number S, related to an arbitrary number N  (equal or greater than 60 and ending in 0 or 5):
Since the twenty five numbers posted in the Table in Step (I) should increase consecutively, let's assume S as the lowest number (or the number starting the sequence) with respect to the number N and occupying the chart location T31. Moreover, as the sum of numbers along any row (or a column) in the Table (chart) should equal N, the sum of all the numbers on the 5 x 5 chart (Table) in 5 rows (or columns) will equal 5*N. The corresponding mathematical expression for S for a given N  (ending in 0 or 5) in terms of the twenty five consecutively increasing constituent numbers will be as, S = N/5 - 12

(III) The basic sequence of numbers on the chart (Table) corresponding to the smallest N:

Next, let's fill the locations / boxes (T11, T12, ...., T55) for the 5 x 5 chart (Table) in Step (I) with successively increasing twenty five numbers (0, 1, 2, ...,24) in the order (sequence) indicated below, starting with 0 (corresponding to the number S in the case of smallest N = 60) in T31 location, resulting in the following chart with numbers along any row, or column, or diagonal adding to 60.

14, 15, 21, 2, 8
7, 13, 19, 20, 1
0, 6, 12, 18, 24
23, 4, 5, 11, 17
16, 22, 3, 9, 10

Results and Discussion:

Finally, given any number N (greater than 60 and ending in 0 or 5), we calculate the lowest number S (as S = N / 5 - 12)  and complete the Table / chart in Step (III): first place the value of  S in T31 location and then successively increase it (as S + 1, S +2, ....., S+24) to fill the remaining locations (in the order indicated in Step III: T45, T54, T13, T22, T23, T32, T41, T55, T14, T15, T24, T33, T42, T51, T52, T11, T25, T34, T43, T44, T53, T12, T21, T35). This results in twenty five consecutively increasing numbers filling the 5 x 5 chart (Table), the addition of which along any row, or column, or diagonal leads to the answer equaling the original number (N), as demonstrated in the examples (1 -  4) in the following.

(1) N = 205

Table of constituent numbers (by using S = N/5 - 12 = 29 in step III):

43, 44, 50, 31, 37
36, 42, 48, 49, 30
29, 35, 41, 47, 53
52, 33, 34, 40, 46
45, 51, 32, 38, 39

(2) N = 305

Table of constituent numbers (by using S = N/5 - 12 = 49 in step III):

63, 64, 70, 51, 57
56, 62, 68, 69, 50
49, 55, 61, 67, 73
72, 53, 54, 60, 66
65, 71, 52, 58, 59

(3)  N = 320

Table of constituent numbers (by using S = N/5 - 12 = 52 in step III):

66, 67, 73, 54, 60
59, 65, 71, 72, 53
52, 58, 64, 70, 76
75, 56, 57, 63, 69
68, 74, 55, 61, 62

(4) N =  1630

Table of constituent numbers (by using S = N/5 - 12 = 314 in step III):

328, 329, 335, 316, 322
321, 327, 333, 334, 315
314, 320, 326, 332, 338
337, 318, 319, 325, 331
330, 336, 317, 323, 324

: Subhash C. Sharma
(Feb. 12, 2019)

http://creative.sulekha.com/an-enchanting-property-of-numbers-greater-than-60-and-ending-in-0-or-5_636151_blog

Seva Lamberdar

Posts : 5764
Join date : 2012-11-29

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