This game has been around for over a year but I only recently started playing, mostly because the game was only available for the US market and I did not went for the Japanese version. A friend was actually looking for a game to play on Android and I suggested this game. I also went ahead and download the iOS version from the US iTunes App Store. It turns out that the game was very good and also addictive!
A lot of people are calling this a Bejewelled-meets-Pokemon. I think the game resembles Pokemon in the aspect of the monsters (called cards in-game but this is not a card game) but the match-3 gameplay is actually more like solving a 2D Rubik Cube. I think the challenge of solving the puzzles (as you will deal more damage when you match many combos) is one of the things that attracts players to the game, as well as frequent events where you can get more magic stones, the in-game currency, and limited chances to get more monsters.
The game is free and supported by in-app purchases but there are many people who played without spending anything. This game is the only one other than Clash of Clans that is making millions per day and interestingly, it was reported that both game companies are "combining" (GungHo of PAD acquiring Supercell of Clash of Clans) A 3DS version of PAD is also coming soon, launching in Japan in December.
I like this game a lot and am still playing it. I've also done a lot of research on the game and will be posting them on upcoming posts.
iTunes App Store link (English): Puzzle & Dragons
iTunes App Store link (JP App Store): Puzzle & Dragons (Japanese)
Android Google Play Store (For US region only): Puzzle & Dragons
(You can get the installer apk file directly from this forum thread)
Note: If you the above iTunes link takes you to GungHo's page with a Korean-named PAD, that is not the full game but instead allows you to try out the game in a short time attack version. The game is in Korean though but there are not much options so you will not get lost!