Another remarkable contribution by Lee Sallows:

(Thanks, Lee!)

Another amazing contribution by Lee Sallows:

“The picture above shows a 4×4 geomagic square, which is to say a magic square using geometrical shapes that can be fitted together so as to form an identical target shape, in this case a 4×6 rectangle, rather than numbers adding to a constant sum. In addition, the square is also panmagic, meaning that besides the usual 4 rows, 4 columns, and both main diagonals, the shapes occupying each of the so-called ‘broken’ diagonals, afkn, dejo, cfip, bglm, chin, belo, are also able to tile the rectangle. Lastly, the 4 shapes contained in the corner cells of the four embedded 3×3 sub-squares, acik, bdjl, fhnp, egmo, are also ‘magic’, bringing the total number of target-tiling shape sets to 20, a small improvement over the 16 achieved by a panmagic-only square. With that said, it is worth noting that 4×4 geomagic squares have been found achieving target-tiling scores as high as 48.”

Click the image to expand it. Thanks, Lee!

A late, beautiful contribution by Lee Sallows:

See “Extra Magic” Realized.

(Thanks, Lee!)

From Lee Sallows:

The drawing at left above shows an unusual type of 3×3 geomagic square, being one in which the set of four pieces occupying each of the square’s nine 2×2 subsquares can be assembled so as to tile a 4×8 rectangle. The full set of subsquares become more apparent when it is understood that the square is to be viewed as if drawn on a torus, in which case its left-hand and right-hand edges will coincide, as will its upper and lower edges. In an earlier attempt at producing such a square several of the the pieces used were disjoint, or broken into separated fragments. Here, however, the pieces used are nine intact octominoes.

(Thanks, Lee!)

From Lee Sallows:

A recent contribution to Futility Closet showed an atypical type of 3×3 geometric magic square in which the 4 pieces occupying each of its nine 2×2 subsquares are able to tile the same rectangle. A different square with the same property is seen in the figure here shown, where the nine tiled rectangles appear at right.

As in the earlier example, the square is to be interpreted as if drawn on a torus, the relations among its peripheral cells then being the same as those that result if the square is surrounded with copies of itself, as seen in the following figure showing four such copies, one in each quadrant:

The figure makes it easier to identify the different 2 × 2 subsquares, exactly nine distinct examples of which can be identified. A brief commentary on the square pointed out that the number of ‘magic’ conditions it satisfies is one greater than the eight conditions demanded by a conventional 3 × 3 magic square. Hence the title of the piece, ‘Extra Magic.’

It was while perusing this diagram that an alternative division of the cells into sets of 4 suggested itself. Instead of 2 × 2 subsquares, consider the four cells defined by a cross that can be centered on any chosen cell. The above figure shows a yellow-shaded example, along with a rectangle tiled by its four associated shapes. It is interesting to note that, as before, there are just nine distinct crosses of this kind to be found in a 3 × 3 square. An obvious question thereby prompted was whether or not a new 3 × 3 magic square could be found based upon such crosses rather than 2×2 subsquares? The answer turned out to be yes, but in the process of scrutinizing an initial specimen I noticed that although it embodied nine cross-based sets of 4 rectangle-tiling pieces, as required, it also included a couple of additional rectangle-tiling sets contained within 2 × 2 subsquares. Clearly the maximum number of such surplus sets would be nine, one for each cross, but could a specimen showing nine cross-based and nine subsquare-based rectangle-tiling sets really exist? I lost no time in seeking an answer.

Regrettably, I was unable to find one. However, the figure below shows a close approach to perfection. It is the same 3 × 3 square with which we started, but now shown alongside no less than eight additional rectangles, each of them tiled with a set of 4 pieces belonging to a cross. Note that the missing rectangle is the one belonging to the non-magic central cross, a show of symmetry that seems appropriate.

So whereas a 3 × 3 magic square, numerical or geometric, satisfies at least 8 separate conditions (3 rows + 3 columns + 2 diagonals), the square here shown satisfies no less than eight more.

(Thanks, Lee.)

From Lee Sallows:

The traditional magic square is a square array of n×n distinct numbers, their magical property being that the sum of the n numbers occupying each row, column, and diagonal is the same. A variation on this theme that I introduced in 2011 is the geometric magic square in which distinct geometrical figures (usually planar shapes) occupy the cells of the array rather than numbers. The magical property enjoyed by such an array is then that the n shapes making up each row, column, and diagonal can be fitted together as in a jigsaw puzzle so as to yield (i.e. tile) a new compound shape that is the same in each case.

Beyond ‘ordinary’ geometric magic squares, it turns out that the combinative properties of shapes are such as to enable ‘magical’ constructions that are denied to analogous structures using numbers. For example, at left in the figure above is seen a 3×3 square of a kind that cannot be realized using distinct numbers rather than shapes. Note first that the square is to be understood as ‘toroidally-connected’, which is to say, as if inscribed on a torus. Its left-hand edge is then to be interpreted as adjacent to its right-hand edge and its top edge adjacent to its bottom edge. Its magical property is then that the four pieces contained within any 2×2 subsquare can be assembled to produce an identical shape, in this case a rectangle of size 4×5. In all there are nine such subsquares to be found in the square, as seen (again in a square) at right. Note that three of the pieces are disjoint, my attempts to produce a similar solution using nine unbroken pieces having failed. So whereas a 3×3 magic square, numerical or geometric, satisfies at least 8 separate conditions ( 3 rows + 3 columns + 2 diagonals), the square here shown satisfies one more.

(Thanks, Lee!)

A stunning geometric alphamagic square by Lee Sallows. The 3 × 3 grid is a familiar magic square in which each number is spelled out: The first cell contains the number 25, the second 2, and so on. Interpreted in this way, each row, column, and long diagonal sums to 45.

But there’s more: The English name of the number in each cell has been arranged onto a distinctive tile, such that the three tiles in any row, column, or long diagonal can be combined to form the same 21-cell figure, as shown. (Shapes with dotted outlines have been turned over.)

And yet more: Count the number of letters in each of the number names (or, equivalently, count the number of cells that make up each tile). So, for example, TWENTY-FIVE has 10 letters, so replace the TWENTYFIVE tile with the number 10. Similarly, replace TWO with 3, EIGHTEEN with 8, and so on. This produces another magic square:

10  3  8
 5  7  9
 6 11  4

Each row, column, and long diagonal totals 21.