When do we begin?

Logic Level 4

I have two hourglasses. One measured 9 minutes, and the other measured 22 minutes. If I want to measure 33 minutes, what is the minimum number of times I need to turn the hourglasses?

Details and Assumptions :

As an explicit example, suppose I want to measure 4 minutes. Then I will have to turn both hourglases at the start. At the 9th minute, I will flip the 9-minute hourglass again. And when the same hourglass is finished, which is at the 18 minute mark, I would start the interval of the 4 minute interval. When the 22-minute hourglass is finished. I will have measured $22-18=4$ minutes interval. Which gives a total of 3 turns of the hourglass, 2 from the 9-minute hourglass, and 1 from the 22-minute hourglass.

Bonus : Prove that we can measure all the integral minute interval using these two hourglasses alone.

Image Credit: Flickr Alicia Llop .

The answer is 14.

3 solutions

汶良林
Aug 10, 2015

9x + 22y = 33

9x = 33 - 22y

x = (33 - 22y)/9 = 3 - 3y + (6 + 5y)/9

y = -3, x = 11

9×11 - 22×3 = 33

11 + 3 = 14

David Ortiz
Jul 2, 2015

Am I weird for just writing out multiples of 9 and 22 and then seeing which were 33 away from each other and then figuring out the corresponding number of hourglass turns (i.e. 99 and 66 => 11 and 3 turns = 14 turns)?

Nope. This is the correct approach to solve all 2 hourglass problems. +1

Pi Han Goh - 5 years, 11 months ago

I think it should be explained more in order to understand better the possible intervals of time that can be measured but maybe it's just me.

I mean that there should be firstly understood the way the hourglasses can be moved so that there is after a sequence of moves some result generally by considering all the possible cases in a more or less synthetic way and afterwards since there is an understanding a priori of the possible moves to speak about what is the smallest possible result relating on some expression.

A A - 5 years, 1 month ago

Feel free to explain it more in detail.

Pi Han Goh - 5 years, 1 month ago

@Pi Han Goh – Hmmmnmm , what about posting a detailed solution to show what I mean ?

Anyways , I'll post too later about the large "Tic tac toe 4" , I think.

A A - 5 years, 1 month ago

@A A – Sure sure, take your time!

Pi Han Goh - 5 years, 1 month ago

@Pi Han Goh – Btw I think there is a solution posted by Charlz Charlizard in just 11 turns if they are counted right so he demans a little bit of attention.

I looked at it and seems right to me , as long as you are allowed to double turn hourglasses even without the sand from one half of the hourglass being completely exhausted. If you have problem with the format let me explain. In short (22-0) (9-0) , (13-9) (0-9) means that the hourglass with 22 minutes has 22 minutes in the upper half and 0 in the lower.

A A - 5 years, 1 month ago

@A A – Yep, that's correct! =D

Pi Han Goh - 5 years, 1 month ago

@Pi Han Goh – So , you should modify the answer.

I started the solution too but it's too long and maybe will be posted in other day.

A A - 5 years, 1 month ago

@A A – Wait, on second viewing, his solution looks erroneous. I'll get back to it later.

Pi Han Goh - 5 years, 1 month ago

@Pi Han Goh – Ok then and just to know ou make me a little sorry for always correcting one or other problem of yours and I hope you don't mind.

A A - 5 years, 1 month ago

@A A – What? Don't ever apologize! We're all learning here!

Pi Han Goh - 5 years, 1 month ago

@Pi Han Goh – Not just here but everywhere and anyone and anyways you have great character , almost philosophical I would say but it can be said that since there can be possible such errors the understanding of how those hourglasses work , the a priori way they behave isn't completely to say the least done and therefore a general understanding is required if possible.

A A - 5 years, 1 month ago

@Pi Han Goh – Yes , I will indeed. I'll try to post later a solution here also too.

A A - 5 years, 1 month ago

Praful Jain
Jun 20, 2015

We can have 9 min hour glass run for 72 min meanwhile 22 min hour glass will run for 66 min .so when 22 min come to a end at 66 min it will take 6 min more for the 9 min to end after that we can start 9 min for 27 min which is equal to 33 min so no of times are (8+3+3) which equals to 14

All integral number of minutes are possible because 22 and 9 are coprime.

Lee Isaac - 5 years, 11 months ago

2(22-0)(9-0),(13-9)(0-9)+1(13-9)(9-0),(4-18)(0-9)+1(4-18)(9-0),(0-22)(5-4)+(start counting 33 minutes)1(22-0)(5-4),(17-5)(0-9)(5 minutes passed)+2(5-17)(9-0),(0-22)(4-5)(10 minutes passed)+2(22-0)(5-4),(17-5)(0-9)(15 minutes passed)+1(17-5)(9-0),(8-14)(0-9)(24 minutes passed)+1(8-14)(9-0),(0-22)(1-8)(32 minutes passed) and 1 minutes remain in 9 minutes measuring hourglass making it 33.. So total number of turns =2+1+1+1+2+2+1+1=11 turns.. FORMAT=number of turns(flipped state)(state when next flipped is to be done)

Charlz Charlizard - 5 years, 11 months ago

Nice trick to count 5 minutes by achieving firstly 5 and then counting it to make 15 and then 24 and then 32 by calculating the difference between the number added in the last and hourglass with 9 m and I for one think you are right though maybe you should explain more the format cause it's unclear and maybe intimidates readers. If it is allowed to flip hourglass before they end numbering their minutes then indeed your solution should be considered correct or at least so I think.

A A - 5 years, 1 month ago