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Assuming water (no pressure) water will run from 1 to 3 first as the outlet from 1 to 3 is lower than 1 to 2.
The outlet from 3 to 4 is lower than from 3 to 5 thus 4 will fill first. Frankly Scarlett
Actually it is 5. I missed the blockage on 4. Ouch Red Gully is right. All of 3 is blocked ! Chore Time!
 
#6 will fill up completely first, cause it's the smallest, taking the very bottom of cup #2. But only if the water it turned on full... filling cup #1 to its higher overflow port. The answer can effectively be multiple answers...depending on the rate of flow out of the brass spigot. If it's a drip of flow the answer is cup # 4, but with good high flow rate...cup #6 will fill fist and win. If the water is turn on full blast Cup#1 winds. The real answer is... it depends...we need to know that flow rate of the spigot.
 

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