PROBLEM SOLVED!
I downloaded uninstall_flash_player.exe from http://fpdownload.macromedia.com/get/flashplayer/current/support/uninstall_flash_player.exe and executed it. Next, I deleted the folder at C:\Windows\SysWOW64\Macromed. Finally, I used regedit.exe to locate and delete instances of Flash and Macromed.
I then successfully installed Flash 16.0 r0 (16.0.0.305). It tested okay at http://www.adobe.com/software/flash/about/, and I was able to play the Planarity game at Planarity.net.
I removed Flash 16.0 r0 (16.0.0.305) using "Uninstall or change a program" at Programs and Features on the Windows Control Panel. After that, I successfully installed Flash 17.0 r0 (17.0.0.134).
I previously tried to install Flash 17.0 r0 (17.0.0.134) without first removing Flash 16.0 r0 (16.0.0.305). I then tried to do that installation after using either Programs and Features on the Windows Control Panel or uninstall_flash_player.exe to remove Flash along with deleting C:\Windows\SysWOW64\Macromed. All those attempts were unsuccessful. Thus, I think I have proven that a failure to remove a prior version of Flash before installing a new version will corrupt Windows to the extent that a manual cleanup of the Windows registry becomes necessary.