Playing with jigsaw puzzles on Shabbos:
Puzzles are made up of various pieces, each piece containing some part of the picture or letter/word that the puzzle desires to portray. Putting a puzzle together on Shabbos touches upon the question of whether it transgresses the prohibition of writing on Shabbos, and taking apart a puzzle raises the question of whether it transgresses the prohibition of erasing on Shabbos. The following is an analysis on this subject:
The general law: It is forbidden to write or erase the letters of a word on Shabbos. Likewise, it is forbidden to draw or erase pictures on Shabbos. It is Rabbinically forbidden to do so even if the letter or picture will not last a long time. It is however disputed as to whether the prohibition applies even in a case that the parts of the letter or picture already exist, and one simply places them near each other, or distances them from each other on a temporary basis. An example of such a case is opening or closing a book that had letters or pictures written/drawn on the side pages of a book. Practically, the custom is to be lenient to allow opening and closing such books on Shabbos [although it should initially be avoided]. The question of whether it is permitted to put together, or take apart jigsaw puzzles, on Shabbos, is dependent on how one understands the above dispute, as will now be explained.
The law by jigsaw puzzles: Some Poskim rule that even according to the lenient opinion, and custom, brought above, it is forbidden to put together, or take apart, jigsaw puzzles which contain pieces that create letters or pictures. Other Poskim however rule that this case follows the same leniency brought above, and hence it is permitted to build or take apart a puzzle on Shabbos even if it creates letters and pictures. Practically, from Admur it is implied like the latter approach, that the leniency applies in this case as well. Nonetheless, as stated above, initially one is to avoid entering the dispute and hence should not build, or take apart such puzzles on Shabbos, although one does not need to protest those who do so. Likewise, one may be lenient to allow children below Bar/Bas Mitzvah to play with such puzzles on Shabbos. If placing the pieces together does not create new letters or pictures, it is permitted according to all to put together or take apart, even if it creates a word, such as if each piece contains a complete letter.
It is disputed whether one may play with jigsaw puzzles on Shabbos. Practically, one is to avoid doing so, although those who are lenient have upon whom to rely. This leniency especially applies to young children.
 Admur 340:4
 Admur 340:10; M”A 340:6; Rambam Shabbos 11:17; Degul Merivava 340; Tehila Ledavid 340:3
 Admur 340:6
 Some Poskim rule it is forbidden to open or close them on Shabbos due to the writing and erasing prohibition. [1st opinion in Admur 340:4; M”A 340:6; Levush 340:4; Menorah Hatehora 340:4] However other Poskim rule it is permitted to close and open [the books]. [2nd opinion in Admur 340:4; Taz 340:2; Shut Rama 119; Maharash Halevi 27] The reason for this lenient opinion is as follows: As since the letters are already written and it is just that they are lacking proximity, this does not contain a writing prohibition. As since it is possible to bring them close together easily without doing any new action they are considered like they are close and standing and one is doing nothing with this proximity. Similarly, it is permitted to open them for this reason and one is not considered like he is erasing them as their writing [remains] intact and it is possible to near them to each other easily and they are [thus] considered already now to be] as if [they were] close. [Admur ibid; Taz ibid; Nishmas Adam 37:2; See Shut Rama ibid for other reasons mentioned; See Piskeiy Teshuvos 340 footnote 97]
 Admur ibid; Kneses Hagedola, brought in M”A 340:6; Tzemach Tzedek in Mishnayos Shabbos 12:4; Tosefes Shabbos 340:9; Birkeiy Yosef 340:5; Mor Uketzia 340; Chayeh Adam 38:5; Levushei Mordechai 1:59; Aruch Hashulchan 340:27; Mishneh Berurah 340:17; Poskim in Kaf Hachaim 340:30 and so concludes Kaf Hachaim ibid; Igros Moshe 2:40; Minchas Yitzchak 7:15; Poskim in Piskeiy Teshuvos 340:10 footnote 98
Other opinions: Some Poskim rule like the stringent opinion, that it is forbidden for the book to be opened or closed. This would likewise apply if there are designs on the edges of the papers. [Avnei Neizer 210]
 Ketzos Hashulchan 144:3; Shaar Hatziyon 340:25 in name of Achronim; Sefer Hachaim; Poskim brought in Piskeiy Teshuvos 340 footnote 98]
 Piskeiy Teshuvos 340:16-footnote 51 [old] 340:19 [new]
 Shalmeiy Yehuda 5:3 in name of Rav Elyashiv; Bris Olam Mocheik 8, Az Nidbaru 5:18; Igros Moshe Yoreh Deah 2:75 regarding the prohibition in placing torn pages together
 The reason: This is Rabbinically forbidden due to the writing prohibition. It is not similar to the allowance of closing a book with letters written on its pages, being that by a book, the pages are bound together and is already considered to be very close. [Az Nidbaru ibid based on 1st reason in Teshuvas Harama ibid] Likewise, here one has intent to create a letter or picture.
 Beir Moshe 6:26; Yesod Yeshurun 1:53; SSH”K 16:23; Mishneh Halachos 6:89 and Beir Moshe 6:125 regarding the allowance to place torn pages.
 The reason: As this is similar to closing a book with writing on its edges which is allowed, as so long as the pieces of the letter were made before Shabbos, and it is merely lacking approximation, there is no writing prohibition relevant.
 See Admur ibid who only records the 2nd reason in Teshuvos Harama, “As since the letters are already written and it is just that they are lacking proximity, this does not contain a writing prohibition. As since it is possible to bring them close together easily without doing any new action they are considered like they are close and standing and one is doing nothing with this proximity.” This is opposed to the 1st reason in Rama ibid who states the allowance is due to that the book is bound and meant for opening and closing. Thus, we see that Admur ibid negated this reason and accepted the reason that applies equally to jigsaw puzzles.
 See Ketzos Hashulchan 144:3
 Piskeiy Teshuvos ibid
 Chayeh Adam; Ketzos Hashulchan 144:10 regarding cases brought there [to sew letters onto a peroches, that if they are weekly sewn they contain no writing prohibition]; Igros Moshe 1:135; Piskeiy Teshuvos 340:7 and 16; Beir Moshe 6:26; Avnei Neizer, brought in Ketzos Hashulchan 144 footnote 4, rules that placing letters near each other has no prohibition.