February 18th, 2019, 02:29 PM  #1 
Newbie Joined: Feb 2019 From: Scotland Posts: 6 Thanks: 0  Number of steps to form 1 fluid oz
Jane wants to have two separate 1 fluid ounce measures of water at the same time. However the only measures she has are for six, ten and fifteen fl oz. Show how this could be done in the smallest number of steps without marking the measures or using any container other than the original large beaker of water. The only steps allowed are filling or emptying a measure or transferring water from one measure to another.

February 18th, 2019, 04:27 PM  #2 
Senior Member Joined: Aug 2012 Posts: 2,311 Thanks: 706 
You can fill the ten, pour it into the fifteen, then fill the six and pour it into the fifteen. Now you have 1 ounce in the six. Now you need a second ounce, but you can no longer use the six? Or if you can, pour the contents of the six (the 1 ounce) into the empty beaker, if you're allowed to empty the beaker. Now repeat the same procedure and you end up with 1 ounce in the beaker and 1 ounce in the six. How one shows that's the minimum solution, I don't know. Can we empty the beaker? You said we can empty the measures, did you mean to exclude emptying the beaker? If you can't empty the beaker you have no place to put that one ounce that won't interfere with getting the second ounce. Unless there's some other way. Last edited by Maschke; February 18th, 2019 at 04:30 PM. 
February 19th, 2019, 02:16 AM  #3  
Newbie Joined: Feb 2019 From: Scotland Posts: 6 Thanks: 0  Quote:
 
February 19th, 2019, 06:22 AM  #4 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,417 Thanks: 1025  
February 19th, 2019, 07:42 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,642 Thanks: 2084 
The rules permit using the beaker, but emptying the beaker isn't a valid step. However, Maschke's procedure can be modified to work, though requiring more steps. After Maschke's first 4 steps, transfer 10 from the 15 to the 10, then empty the 15, transfer all the water in the 10 to the 15, refill the 10 from the beaker, transfer 5 from the 10 to the 15 (filling it), and then empty the 15. You now have 1 in the 6, 5 in the 10 and the 15 is empty. Transfer the 1 in the 6 to the 15, fill the 6, and then transfer 5 from the 6 to the 10 (filling it). There is now 1 in the 6, 10 in the 10 and 1 in the 15, achieving the desired objective after a total of 13 steps. I don't know whether a shorter method is possible. 
February 19th, 2019, 08:01 AM  #6 
Newbie Joined: Jan 2017 From: London Posts: 11 Thanks: 0 
I found a similar question but with 2 "jugs" instead of 3... https://math.stackexchange.com/quest...ofoperations I don't really understand everything but maybe it could be applied for this question? 
February 19th, 2019, 08:11 AM  #7  
Newbie Joined: Feb 2019 From: Scotland Posts: 6 Thanks: 0  Quote:
 
February 19th, 2019, 09:52 AM  #8  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,417 Thanks: 1025  Quote:
If the rules did NOT permit using the beaker, then where would the water come from? You HAVE to use the beaker: it's the source of the water...no? If beaker is used only as a water source, then there is no need to mention it, and the problem would be CLEAR this way: Jane wants to have 2 separate 1 ounce measures of water at the same time. However the only measures she has are for 6, 10 and 15 ounces. So she needs to end up with 2 of these 3 measures containing 1 ounce each. Show how this could be done in the smallest number of steps, without marking the measures or using any other container. The only steps allowed are filling or emptying a measure or transferring water from one measure to another. Assume the kitchen water tap is used as the source of water!  
February 19th, 2019, 12:05 PM  #9 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,417 Thanks: 1025 
I "nervously!" have a 10 step solution (water from kitchen tap!): Code: STEP [15] [10] [6] 0 0 0 1 0 10 0 : fill [10] 2 10 0 0 : [10] to [15] 3 10 0 6 : fill [6] 4 15 0 1 : fill [15] from [6] 5 5 10 1 : fill [10] from [15] 6 0 10 1 : empty [15] 7 10 0 1 : dump [10] in [15] 8 10 1 0 : dump [6] in [10] 9 10 1 6 : fill [6] 10 15 1 1 : fill [15] from [6] 
February 19th, 2019, 06:38 PM  #10 
Global Moderator Joined: Dec 2006 Posts: 20,642 Thanks: 2084 
That works. There's another solution in 10 steps, and 10 seems to be the minimum number of steps.


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