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6.AS.5. RECTANGLE TO A SQUARE OR OTHER RECTANGLE
New section. See comment at 6.AS. The dissection of a 5 x 1 into five pieces which make a square is explicitly covered in 6.AS.1, and the other cases above can be viewed as dissections of 2 x 1, 3 x 1 and 8 x 1. There must be older examples of the general case??
Ozanam Montucla. 1778.

Avec cinq quarrés égaux, en former un seul. Prob. 18 & fig. 123, plate 15, 1778: 297; 1803: 292-293; 1814: 249-250; 1840: 127. 9 pieces. Remarks that any number of squares can be made into a square.

Prob. 19 & fig. 124-126, plate 15 & 16, 1778: 297-301; 1803: 293-296; 1814: 250 253; 1840: 127-129. Dissect a rectangle to a square.

Prob. 20 & fig. 125-126, plate 15 & 16, 1778: 301-302; 1803: 297; 1814: 253; 1840: 129. Dissect a square into 4, 5, 6, etc. parts which form a rectangle.

"Mogul". Proposal [A pretty geometrical problem]. Knowledge 1 (13 Jan 1882) 229, item 184. Dissect a rectangle into a square. Editor's comment in (3 Mar 1882) 380 says only the proposer has given a correct solution but it will be held over.

"Mogul". Mogul's Problem. Knowledge 1 (31 Mar 1882) 483. Gives a general construction, noting that if the ratio of length to width is  2, then it takes two cuts; if the ratio is in the interval (2, 5], it takes three cuts; if the ratio is in (5, 10], it takes four cuts; if the ratio is in (10, 17], it takes five cuts. In general if the ratio is in (n²+1, (n+1)²+1], it takes n+2 cuts.

Richard A. Proctor. Our puzzles; Knowledge 10 (Nov 1886) 9 & (Dec 1886) 39-40 & Solution of puzzles; Knowledge 10 (Jan 1887) 60-61. "Puzzle XII. Given a rectangular carpet of any shape and size to divide it with the fewest possible cuts so as to fit a rectangular floor of equal size but of any shape." He says this was previously given and solved by "Mogul". Solution notes that this is not the problem posed by "Mogul" and that the shape of the second rectangle is assumed as given. He distinguishes between the cases where the actual second rectangular area is given and where only its shape is given. Gives some solutions, remarking that more cuts may be needed if either rectangle is very long. Poses similar problems for a parallelogram.

Tom Tit, vol. 3. 1893. Rectangle changé en carré. en deux coups de ciseaux, pp. 175-176. = K, no. 24: By two cuts to change a rectangle into a square, pp. 64-65. Consider a square ABCD of side one. If you draw AA' at angle α to AB and then drop BE perpendicular to AA', the resulting three pieces make a rectangle of size sin α by csc α, where α must be  45⁰, so the rectangle cannot be more than twice as long as it is wide. If one starts with such a rectangle ABCD, where AB is the length, then one draws AA' so that DA' is the geometric mean of AB and AB - AD. Dropping CE perpendicular to AA' gives the second cut.

Dudeney. Perplexities column, no. 109: A cutting-out puzzle. Strand Magazine 45 (No. 265) (Jan 1913) 113 & (No. 266) (Feb 1913) 238. c= AM, prob. 153 -- A cutting-out puzzle, pp. 37 & 172. Cut a 5 x 1 into four pieces to make a square. AM states the generalized form: if length/breadth is in [(n+1)², n²), then it can be done with n+2 pieces, of which n-1 are rectangles of the same breadth but having the desired length. The cases 1 x (n+1)² are exceptional in that one of pieces vanishes, so only n+1 pieces are needed. He doesn't describe this fully and I think one can change the interval above to ((n+1)², n²].

Anonymous. Two dissection problems, no. 1. Eureka 13 (Oct 1950) 6 & 14 (Oct 1951) 23. An n-step is formed by n lines of unit squares of lengths 1, 2, ..., n, with all lines aligned at one end. Hence a 1-step is a unit square, a 2-step is an L-tromino and an n step is what is left when an (n-1)-step is removed from a corner of an n x n square. Show any n-step can be cut into four pieces to make a square, with three pieces in one case. Cut parallel to a long side at distance (n+1)/2 from it. The small piece can be rotated 180^o about a corner to make an n x (n+1)/2 rectangle. Dudeney's method cuts this into three pieces which make a square, and the cuts do not cut the small part, so we can do this with a total of four pieces. When n = 8, the rectangles is 8 x 9/2, which is similar to 16 x 9 which can be cut into two pieces by a staircase cut, so the problem can be done with a total of three pieces. A little calculation shows this is the only case where n x (n+1)/2 is similar to k² x (k-1)².

Lorraine Mottershead. Investigations in Mathematics. Blackwell, Oxford, 1985. P. 105. Dissect a 2 x 5 rectangle into four pieces that make a square.