Amazing Alex’s performance will be closely watched after the firm’s earlier series racked up more than one billion downloads.
Players must arrange objects on screen to create a chain reaction in which each piece of the puzzle has an effect on another until the goal is achieved.
Rovio might find it hard to repeat its previous success.
The launch was marred by a technical problem that caused error messages saying “this game is not recognized by the Game Center” to repeatedly appear on some users’ Apple devices.
Rovio said the problem was “frustrating” adding that it was trying to resolve the problem “as soon as possible” and suggested affected users should try to reboot their devices for a “temporary fix”.
Unlike Rovio’s other games, Amazing Alex began its life outside the Finnish firm.
It was originally known as Casey’s Contraptions, a physics-based game in which players had to help the lead character free his toys.
It was released in 2011 by Noel Llopis, a California-based independent game developer, and the Washington-headquartered games studio Mystery Coconut.
It received positive reviews which compared the tasks involved to the outlandish machines drawn by the cartoonist Heath Robinson and the ad-hoc inventions of the 1980s television show MacGyver.
However, the title never became a hit and the developers sold their intellectual property rights to Rovio earlier this year, following which it was removed from Apple’s iOS store.
Rovio has given the graphics and gameplay a tweak and has now re-released it on both the iOS and Android platforms.
The firm’s Twitter feed confirmed that versions for PCs, Macs and Windows Phone handsets would follow.
The company has tried to generate excitement for the release by featuring a prominent tease in its Angry Birds Seasons game which it offered as a free download to iPhone and iPad owners for a limited time.
However, one mobile games analyst said it was too soon to know if the move would pay off.
“It’s hard to know if they can bring their audience with them,” said Jack Kent from IHS Screen Digest.
“If you look at a company in the wider games business like Zynga, they have used core brands like CityVille to add players to their next games and have had some success in doing that.
“But on mobile it’s not as clear that Rovio will be able to do that, especially from a game like Angry Birds that doesn’t have the same social connections – it’s a casual game but you’re not necessarily connected via a wider social network.”
Rovio’s financial report for 2011 revealed it generated 75.4 million Euros ($92.3 million) from sales last year.
It said about 30% of that sum was derived from its own spin-off merchandise and licenses to third-parties for products including toys, clothes and themed play parks.
Jack Kent said Rovio’s expertise in this area meant similar efforts were likely to follow Amazing Alex if it proved popular.
But Rovio is not solely focusing its efforts on the new title.
It has told the gaming website IGN that it had teamed up with the publisher Activision to release its first three Angry Birds titles as a bundled package for the Xbox 360, Playstation 3 and Nintendo 3DS later this year.
It said the versions for Sony and Microsoft’s devices could be controlled via the Move and Kinect motion sensors.
But they will face competition from other titles including Wreckateer which also plan to feature a similar dynamic.
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