Problems & Puzzles: Puzzles

Puzzle 263.  MagicAntiMagic Squares

The past Monday, (April 5, 2004) Rodolfo Kurchan sent to me by email the following nice 5x5 antimagic square:

 59 7 8 24 22 2 63 4 16 9 14 21 64 25 11 13 15 5 69 6 12 17 10 23 68 18 20 3 1 19 61 60 67 66 62 70 65

The stunning feature of this antimagic 5x5 square is that contains an embedded centered nut of a 3x3 magic square (numbers in red color); that is to say here we have a magic square inside an antimagic square! A beauty example of an object that contains within its contrary.

Perhaps the beauty of kind of this objects is excuse enough in order to keep away the prime numbers for this puzzle.

Questions:

1. Can you get another antimagic solution of the same size (5x5) using a distinct magic (3x3) square?

2. Can you get an antimagic 5x5 containing an eccentric (non-centered) magic 3x3

3. Can you get a larger example (i.e. an antimagic 6x6 containing a magic 4x4)?

4. Can you get the opposite concept: a magic square containing inside an antimagic square (*)

_______

(*) It has been shown that no antimagic square of order less than 4 can exist; then, the minimal example of this kind of objects could be an magic square 6x6 containing a centered antimagic square 4x4.

Solution:

I feel my self really happy because the readers of my pages are more daring than I suppose. While I was thinking that this puzzle was hard enough in order to add the primality condition of the numbers used, J.C. Rosa got a solutions to Question 4 using only primes! Here is what he wrote:

It is possible to find one 3x3 antimagic square with prime
numbers ( see Won plate 132 ) and particularly
for your puzzle 263 I have found this :

The following antimagic square is composed of nine primes
with its eight sums in arithmetic progression (step 2 ). The sums
go from 443 up to 457:

101    113    233
293    151     13
59    191    199

(note that the central number is a palprime )

And now the same antimagic square embedded in a 5x5 magic square:

29     157    277    263      23
229     101    113    233      73
11     293    151     13     281
197      59    191    199     103
283     139     17     41     269

magic sum = 749

(note that this puzzle is the puzzle 263 and 263 is inside this
square)

***

do you want an 3x3 prime antimagic square
embedded in a 5x5 prime magic square with a prime magic sum ?

Here it is ::

83      43     139       23      101
151     29    113       89         7
41     149     79       13     107
53      59      47      127     103
61     109     11     137       71

For the 3x3 antimagic square the sums go from 227 up to 241 ( step 2). For the 5x5 magic square the magic sum is 389 (prime )

***

J. C. Rosa also contributes to Q1:

About the question 1 of the puzzle 263 there are a lot of different
solutions . Here are two examples with the numbers from 1 to 25
and one example with only 25 prime numbers ( unfortunately they are
not consecutive ).See below. Now I hope to find a bridge between the
question 1 of the puzzle 263 and the question 2 of the puzzle 264 :
A 3x3 magic square embedded in an 5x5 antimagic square composed
only 25 consecutive primes....

5     7     20     18     10
17     2     23      14     6
4     25    13      1      21
19    12     3       24    11
16    22     8       9      15

Magic sum=39 . The sums go from 59 to 70

xxxxxxxxx

5    14     18     17     8
15     2     21      13     10
4     23    12      1      25
24    11     3       22     7
20    16     9       6      19

Magic sum=36 . The sums go from 59 to 70

xxxxxxxxxxxxx

67     31       23      107      61
7      17       89       71      109
41     113      59         5        83
127    47       29       101      3
53      97       103      13       43

Magic sum=177    The sums go from 287 to 309 (step 2)

***

J. C. Rosa contribution to Q2 arrived the 12/6/04:

About the question 2 of the puzzle 263 I have found
many solutions. Here are only three examples
(the magic squares are in bold letters at the top left corner )

a) with the numbers from 1 up to 25 :

2      21     13       19        5
23     12       1       18         7
11     3       22       10       20
15      4      25        9       17
16     24     8          6       14

Magic sum=36 . The sums of the antimagic 5x5 go from 59 up to 70.

b) with 25 prime numbers (they are not consecutive ) :

17      89     71       109     43
113     59      5          53    103
47     29     101        7      157
37      3      151      137     19
131    163     11       31       13

Magic sum=117 . The sums of the antimagic 5x5 go from 327 up to 349 (step 2).

c) (the best till the end ! ) with 25 CONSECUTIVE PRIME NUMBERS :

(moreover this example is a solution of the question 2 of the puzzle 264 )

41      89     83       79      37
113      71     29       73      61
59     53     101      31      97
109    103     47      23      43
17     19      67     127      107

Magic sum=213 . The sums of the antimagic 5x5 go from 325 up to 347 (step 2).

***

For the question 3, Rodolfo Kurchan wrote (Feb 18, 2005):

In 2005 I found an antimagic 6x6 square that contains in the center a 4x4 magic square:

 108 1 36 34 33 2 3 109 35 26 13 12 23 6 115 27 15 20 21 18 5 106 10 19 16 17 22 30 114 9 14 25 24 11 29 112 31 7 8 4 28 32 110 113 117 116 111 104 105 107

***

Anurag Sahay had previously sent (Jan 2005) the following solutions to Q3:

> Some solutions for Q3:
>
> 5 32 29 3 6 30
> 27 12 19 18 25 8
> 36 26 24 11 13 7
> 9 15 17 22 20 31
> 34 21 14 23 16 2
> 4 10 1 35 33 28
>
> 7 29 34 3 9 28
> 30 12 19 18 25 10
> 6 26 24 11 13 31
> 35 15 17 22 20 8
> 33 21 14 23 16 2
> 5 1 4 36 32 27
>
> 6 7 34 29 32 2
> 33 12 19 18 25 10
> 4 26 24 11 13 36
> 8 15 17 22 20 27
> 30 21 14 23 16 1
> 31 35 5 3 9 28
>
> 4 3 34 36 31 2
> 28 11 25 12 26 9
> 1 22 20 17 15 30
> 8 23 13 24 14 35
> 33 18 16 21 1

***

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