## Friday, January 6, 2012

### A 9-minute steak

This is a famous tricky puzzle wrapped in a new story.

You are a cook in a restaurant. A regular picky customer orders a steak that should be done in exactly 9 minutes. How can you measure these 9 minutes if the only precise clocks you have are a four minute hourglass, and a seven minute hourglass?

Top image by kamums, distributed under CCL.

Your thoughts and ideas accepted any time until midnight on Sunday Janury 8th(EST), on our Family Puzzle Marathon. They will be hidden till then and everyone who submitted something reasonable will get a puzzle point.

Kim said...

Time 0: Start both timers and run them for 4 minutes.

Time 4: The four minute timer has run out, the 7 minute timer has three minutes left in it. Turn over the 4 minute timer, and let the 7 minute timer keep going.

Time 7: The 7 minute timer will run out, and the 4 minute timer will have 1 minute left on it.

Turn the 7 minute timer over and let it run until the 4 minute timer finishes running out. This will be at 8 minutes.

Time 8: Now turn over the 7 minute timer, which will have 1 minute of sand in it, and run it until it runs out.

Definitely worth getting a digital timer instead!

Anonymous said...

Here’s my solution to cooking the perfect 9 minute tofu (since I am a vegetarian!)

You take both hourglasses and flip them over
After 4 minutes the top and bottom of each show:
3 top – 4 bottom and 0 top - 4 bottom
Flip the 4 minute one again as soon as it runs out and let the 7 minute one continue in the same direction until it runs out (after 3 minutes) the hourglasses now show
0 top – 7 bottom and 1 top – 3 bottom and you are now at 7 minutes
Flip the big one and let the 4 minute one continue until it runs out (after one minute) and now they each show
6 top – 1 bottom and 0 top - 4 bottom and you are at 8 minutes
Flip the 7 minute hourglass again, reversing the flow and when it runs out you have reached 9 minutes exactly.

Thanks for the great puzzles!

Submitted by Eirinn C.

Wang said...

There's lots of preparation beforehand... I'm not so sure that the picky customer would actually want to wait but:

Turn the 4 minute hour glass and 7 minute hour glass over at the same time.

4 minute hour glass (0 minutes left)
7 minute hour glass (3 minutes left)

Once the 4 minute hour glass is done, flip that over.

4 minute hour glass (4 minutes left)
7 minute hour glass (3 minutes left)

Once the 3 minutes are up in the 7 minute hour glass, flip the 7 minute hour glass over.

4 minute hour glass (1 minute left)
7 minute hour glass (7 minutes left)

Once 1 minute has passed, flip the 4 minute hour glass over

4 minute hour glass (4 minutes left)
7 minute hour glass (6 minutes left)

Once the 4 minutes has passed, start cooking the steak.

4 minute hour glass (0 minutes)
7 minute hour glass (2 minutes)

Flip the 7 minute hour glass when the 2 minutes are up and continue to cook the steak until done in exactly 9 minutes (2 + 7 minutes).

kj said...

Start both timers simultaneously. As soon as the four minute hourglass runs out, flip it back over.

At this point the total time elapsed is 4 minutes and the four minute timer has 4 minutes left, and the seven minute timer has 3 minutes left.

As soon as the seven minute hourglass runs out, flip it back over.

At this point the total time elapsed is 7 minutes, and the four minute timer has 1 minute left, and the seven minute timer has 7 minutes left.

As soon as the four minute hourglass runs out, reverse the seven minute timer. (You may ignore the four minute hourglass now.)

At this point the total time elapsed is 8 minutes, and the seven minute timer has 1 minute left.

When the seven minute hourglass runs out, the steak will be done.

Ilya said...

For brevity, let's call the hourglass timers T7 and T4. Here is the process:
Start T7 and T4 simultaneously.
When T4 finishes, T7 has exactly 3 minutes left, so start cooking the stake, also start T4 again.
When T7 finishes, we are 3 minutes into cooking. Bottom part of T4 has 3 minutes worth of sand, flip it. Start T7 again.
When T4 finishes, we are 6 minutes into cooking. Bottom part of T7 has 3 minutes worth of sand, flip it.
When T7 finishes, the stake has been cooking for 9 minutes. stop.

SteveGoodman18 said...

This can be done, if we run the timers for a bit before we actually start cooking the steak.

Start both timers. 4 minutes later, the 4-minute timer has expired, so flip it. 3 minutes later, the 7-minute timer has now elapsed and the 4-minute time has 1 minute left in it. This is when we begin the steak.

Now simply cook the steak for the remaining 1 minute on the 4-minute timer, plus 2 more complete cycles of that timer for a total of 9 minutes.

Prep time: 7 Minutes
Cook time: 9 Minutes

Anonymous said...

Okay. I had my 8th grade math class work on this. They came up with 2 solutions. The 1st solution is to run the 7 minute timer then run the 4 minute time and stop the 4 minute time half way. The 2nd way was to record minutes on the hourglasses at home before you bring them to work. That way you can measure the minutes as they go by. - TMS 8th grade

Jerome said...

I can't think (at the moment) of any way to do this except the following.

Before the customer comes in, do the following

Set both timers to zero. This means that if they were not set so they could run to their full capacity, make them so they can.

Time the timers so they can run to empty. Do this 3 times with the 4 minute timer.

As the 4 minute timer is doing its thing, when the 7 minute timer runs out, start it again.

When the 4 minute timer runs out for the third time, the seven minute timer still has 2 minutes to go.

Lay the seven minute timer on its side until you start the 4 minute timer again.

When the seven minute timer runs out, lay the 4 minute timer on its side where it has been stopped at its 2 minute point.

When the customer comes in, start the 7 minute timer. When it runs out, run the 4 minute timer which is now set for a 2 minute run.

Total: 9 minutes.

If there is any other way, I'll have to wait until Monday, or have a brain wave infusion that currently is not happening.

anne-marie said...

I turn both hourglasses at the same time.
When the 4 minutes hourglass timing is passed, there are still 3 minutes in the other one.
I turn the four minutes one and when the time in the seven minutes hourglass is passed, there is still one minute in the four minutes hourglass.
I turn the seven minutes one and when the one minute is passed in the four minutes hourglass, I turn back the seven minutes hourglass to get one more minute and reach the nine minutes.

anne-marie said...

I turn both hourglasses at the same time.
When the 4 minutes hourglass timing is passed, there are still 3 minutes in the other one.
I turn the four minutes one and when the time in the seven minutes hourglass is passed, there is still one minute in the four minutes hourglass.
I turn the seven minutes one and when the one minute is passed in the four minutes hourglass, I turn back the seven minutes hourglass to get one more minute and reach the nine minutes.

Anonymous said...

Start the 4minute and the 7minute simultaneously.
Immediately restart the 4minute when its done.
Start cooking the steak when the 7minute is done.
You now have 1 minute before the 4minute is done.
Cook for 2x with the 4minute and you have 9 minutes total.

7 - 4 start
3 - 0 end

3 - 4 start
0 - 1 end
start cooking
0 - 0 end
4 start
0 end
again
4 start
0 end
Finish.
\$12.99 + tax.

Annie said...

Begin by turning both hourglasses over (both were empty on the top), preheating the grill and seasoning the steak, When the four minute hourglass is complete turn them BOTH over. When the 3 minutes left in the 7 min. hourglass is over, turn the four minute hourglass over ( one minute remaining) and put the steak on the grill. When it is empty turn it again (4 more minutes) and again (4 more minutes) and when that turn is complete take the steak off the grill. It has cooked for 1 min. + 4 min. + 4 min. = 9 minutes.

TracyZ said...

By concidence, my family discussed the original puzzle at great length at a gathering over the holidays. (does that disqualify me to answer this puzzle? - I hope not :))

Here's my favorite answer:
Note: this answer only works if the customer isn't in a hurry to get their steak.

(a) start the 4-minute hourglass and 7-minute hourglass at the same time
(b) when the 4-minute hourglass runs out, start the 4-minute hourglass again right away
(c) when the 7-minute hourglass runs out, there will be 1 minute left on the 4-minute hourglass. Start cooking the steak now.
(d) After the 4-minute hourglass runs out again, use the 4-minute hourglass two more times. When 4 minutes has gone by twice, the steak will have cooked for 9 minutes.

1 minute(c) + 4 minutes(d) + 4 minutes(d) = 9 minutes

This approach requires 16 minutes (9+7) to cook a steak for 9 minutes.

Anonymous said...

Start 4 and 7 simultaneously. When 4 runs out, turn it over and let it run. Next, 7 will run out, turn it over and let it run till 4 completes. At this point, 8 minutes have elapsed. 1 min has elapsed in the 7 min hourglass; turn the 7 and let it run through. When 7 completes, a total of 9 minutes have elapsed. Done! - Venkat Krishnamoorthy

Alin Grin said...

Let both hourglass timers start running at the same time. Everytime one finshes it's run, turn it around to let it start running again. as soon as the 4 minutes hourglass finishes it's third run, you put the steak on and let the 7 minutes hourglass finish it's current 2 minutes run then turn it around for it's third 7 minutes run. when it's done, take the steak off.

Brian Ramelson said...

Prior to starting the steak, let both timers go until the 4 minute timer has completed. Then the 7 minute timer has 3 minutes left. Turn over the 4 minute timer and allow it to go for 3 minutes (ie until the seven minute timer has expired). Now there is only one minute left in the 4 minute timer. Start cooking the steak. After the 1 minute in the 4 minute timer has completed, turn over the 4 minute timer until it has completed and then do that once more. The total amount of time that has passed is 1+4+4 or 9 minutes.

Maria said...

Very interesting. There are more solutions than what I thought. I bet you will be surprised as well.

First of all - a puzzle point for everyone who answered. Anonymous - you have to reveal yourself to get it. Welcome new puzzle solvers and welcome back our old-time master Alin Grinn. We missed you!

You all provided a terrific explanations for each of the solutions. I decided to draw the basic solution that does not require any preparation and takes only 4 steps. Click "back" button after viewing this image to return to the answer page.

Click to see the Fastest Solution

Second solution does steps 1 and 2 in advance, before cooking starts. This leaves us with 1 minute left in the 4-minute hourglass. All you need to do now is start the steak, run this 1 minute and add two more flips of the 4-minute hourglass.

Third solution does 3 steps in advance flipping 4-minute hourglass twice that gives us 12 minutes. Meanwhile two flips of a 7-minute hourglass give us 14 minutes. 14-12=2 minutes left in a 7-minute hourglass. At that point you start cooking. Use 2 minutes that are left, then flip and run to the end 7-minute hourglass and you have a 9-minute steak.

TMS 8th grade said...

I am the 2 anonymous answer above

Maria said...

Thanks, TMS! I see your name signed there. What about 3rd anonymous who didn't sign? :)