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Install on your phone: open this page on the phone, tap WorldWoodshed.apk, and allow installs from your browser when Android asks. Later versions install over this one and keep your progress (they are all signed with your own key).
To carry progress between the phone and your computer: Today → "Your progress on every device" → Send my progress on one, Load a sync file on the other. Loading merges, so practice on either device is kept.