[Update: After I wrote this, I found out that Google's Digital Wellbeing app has a built-in Grayscale feature. Ugh.]
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Read about Digital Wellbeing: https://www.androidpolice.com/2018/09/06/digital-wellbeing-update-adds-grayscale-tile-app-drawer-icon/
Toggling colour / monochrome on an Android phone is a much bigger pain than it needs to be but once the instructions are all in one place, it's worth the five minutes it takes to complete.
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Pay $1 and nstall Tiles 
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Open Tiles and scroll down to Monochromacy. TurnTile visibilityon. Edit Quick Settings and add theMonochromacytile.
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Try clicking Write secure settings permission. It will complain, saying the app doesn't have this authority and you need to grant it using Android Platform Tools.
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Install Android Platform Tools. On a Mac: 
brew cask install android-platform-tools
...for Linux and Windows, follow the instructions on the Tiles support wiki: https://github.com/TilesOrganization/support/wiki/How-to-use-ADB-to-grant-permissions
- Plug your phone into your computer with a USB cable and use ADB to override the secure settings permission:
adb devices
# => this will poke your phone and your phone will ask you to give `USB Debugging` permission to your computer; allow this
adb shell pm grant com.rascarlo.quick.settings.tiles android.permission.WRITE_SECURE_SETTINGS
# => no news is good news; this command has no output if it worked correctly
adb shell dumpsys package com.rascarlo.quick.settings.tiles | grep WRITE_SECURE_SETTINGS
# => you should see `android.permission.WRITE_SECURE_SETTINGS: granted=true`
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Try clicking Write secure settings permissionagain. You'll get the same pop-up but it should toggle the setting this time.
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Try your new Monochromacy quick setting! 
update from 2021 - Google removed the grayscale function from Digital Wellbeing, thank for not taking this down!