The Haggadah tells us that Rebbi Yehuda came up with a mnemonic for remembering the Ten Plagues: Dtzach Adash B'Achav
Why was the mnemonic necessary? It doesn't seem to me to be that difficult to remember the plagues. Moreover, there are many other things for which there is no mnemonic (eg. 12 tribes). Is there a deeper meaning behind Rebbi Yehuda's mnemonic?
Abarbanel writes that, because there's a different order for the plagues in Psalms (78? 105? he's unclear), and one might think it's the chronological order (and the order in the Exodus is not, per אין מוקדם ומאוחר בתורה), Rabi Y'huda gave a mnemonic so one knows the chronological order.
h/t Double AA
There are many explanations of the intent of his divisions. Here is one:
The Maharal points out that the plagues follow a pattern, split into units of 3 - the first of each group (plagues 1, 4, and 7) are preceded by a warning to Pharaoh issued by the Nile. The second of each group (2, 5, and 8) are preceded by a warning issued to Pharaoh while sitting on his throne. The third of each group has no warning.
The Maharal explains that the plagues are in 3 groups, working their way up through creation, and each group within itself increases in intensity. The first group is in the earth itself - water to blood, frogs from the water, and the dust being turned into lice. The first had its warning by the Nile, outside of Pharaoh's seat of power. The second was closer to home, with the warning at the throne itself. The third did not even get a warning.
The second group is on the land - plagues against man and animals, again with the same progression of intensity.
The third group is the sky - hail which came from the sky, locusts which blocked out the heavenly bodies, and darkness which completely blocked out the light of the sun, with the same progression.
The plagues were split into these groups of three to split them into their divisions of which part of existence they targeted.
The final plague was a "grand finale" which also did not get a warning, but was the culmination of the entire display.
The Lubavitcher Rebbe said that this abbreviation answers the question of what Hashem meant when He said that He "mocked" the Egyptians when he rescued B'nei Yisra'el from their land. This is alluded to by the fact that d'tza"ch ada"sh b'acha"v is numerically equivalent to the word asher from the pasuk (Sh'mos 10:1-2):
בֹּא אֶל-פַּרְעֹה כִּי-אֲנִי הִכְבַּדְתִּי אֶת-לִבּוֹ וְאֶת-לֵב עֲבָדָיו לְמַעַן שִׁתִי אֹתֹתַי אֵלֶּה בְּקִרְבּוֹ וּלְמַעַן תְּסַפֵּר בְּאָזְנֵי בִנְךָ וּבֶן-בִּנְךָ אֵת אֲשֶׁר הִתְעַלַּלְתִּי בְּמִצְרַיִם
God's mockery of Egypt therefore consisted of the execution of plagues themselves.
Rav Ya'akov Emden gives a whole slew of deeper meanings afforded by this particular formulation of the plagues' names (Recommended! Check it out here.), and one of them contains a similar idea. He says that R. Y'huda's phrase refers more to the order of plagues than their names, since the order recorded in T'hilim 78 seems somewhat different (example). Therefore, to solidify his contention that the order in which Hashem actually doled them out, R. Y'huda gave this abbreviation, whose numerical value alludes to Hashem's own reference to the affliction of Egypt in the phrase (Sh'mos 15:26)
כָּל הַמַּחֲלָה אֲשֶׁר שַׂמְתִּי בְמִצְרַיִם
Rav Yehuda made a siman for the 10 makkas D’tzach ,A’dash ,B’achav three distinct classes. The Kli Yakar explains why three separate classes. The מהרי"א explains that Paroh argued on three points 1)He didn’t believe in Hashem 2) Even if Hashem existed he created the world but doesn’t get involved. 3) He believed even if Hashem is involved he can’t change nature.,(he also did not believe in a one and only God).
The Kli Yakar explains that this siman that Rav Yehuda made for the makkas answers the three questions Pharaoh had. The first set of makkas(D’Tzach) the first makka ,the passuk says with this you will know I am Hashem. The first three makkas Showed that Hashem exists and furthermore pharaohs false god the water which he worshipped was really nothing at all and it even turned into blood. At the end of this set of makkas which was makka kinim(lice) the sorcerers answered Etzbah Elokim(this is the finger of Hashem) so the first set proved parohs first question wrong.
The Second set A’dash by oirev the first makka of the set Hashem said you will know that I am Hashem and I am B’kerev Ha’aretz(dwells in the earth). This set was to prove Pharaohs second question. These Makkas Hashem distinguished between the Rashim and the Tzaddikim and this proves that Hashem didn’t abandon the humans rather he still is B’kerev Ha’aratz. These makkas, Hashem distinguished between the Jews and the mitzrim it would stop by the borders between Jew and Egyptian.
The Third set B’achav came to answers pharaohs third and final question. The first makka of this set is Barad and the passuk says you will know that there is none like me in the whole land (world). Paraoh believed in many false gods the sun the mazalas(stars) especially the sign of Nissan the lamb, he believed he gained strength from these things. So Barad blocked the sun there was no sun by makka arbeh and by chosech as well. By makkas Bechoros it was at night, and the mazal for the month is the bechor of all mazalos. All the powers which pharaoh believed in at the end really turned against him.
Another answer suggested by several Rishonim in their commentaries to the Hagadah (Rid, Pirush Kadmon, Shiboley Haleket):
The following section of the Hagadah records an argument between several Sages regarding how many plauges the Egyptians were smitten with by the sea. R' Yossi Haglili maintains that they suffered from 50 plauges, R' Eliezer claims it was 200 and R' Akiva holds the number was 250. R' Yossi here comes to hint that in fact all the above opinions are correct, as the total of all three opinions is 500 - which is numerically equivalent to דצ"ך אד"ש באח"ב (minus one, based on the Gematria principle of עם הכולל).
Rabbi Glatstein gave a whole shuir in this exact inyun hear the shuir here.
