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On a table in front of you is a small freezer that is capable of holding seven standard ice cube trays stacked on top of each other. There are no shelves to separate the trays, and if you stack one tray on top of another before the ice cubes in the bottom tray are fully frozen, the top tray will sink into the bottom tray and you will not be able to make full-sized ice cubes. You have an unlimited supply of trays, each of which can make a dozen ice cubes.

If it takes fifteen minutes for the water to freeze completely into an ice cube, what is the maximum number of full-sized ice cubes you can produce in sixty minutes?

As per my calculation, one can do 120 cubes each time. Since it takes 15 minutes to make an ice cube, 120X4=480 would be the answer. However, need some validation on whether the maths is correct.

Edit 1:

Seems like a few of us have presumed we are allowed to break or mutilate the trays, unfortunately, it is not part of the deal.

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  • $\begingroup$ can you pls tell how 120 cubes are possible each time $\endgroup$ – user9174 Jul 26 '17 at 8:47
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    $\begingroup$ Yeah, I'm really struggling to see where 120 comes from. The freezer only fits 7 ice cube trays with 12 each so I'd have thought an upper bound on the number of icecubes would have to be 12*7 = 84 every fifteen minutes. You somehow are making 120 every fifteen minutes? $\endgroup$ – Chris Jul 26 '17 at 8:58
  • $\begingroup$ Fill 4 trays with water and turn the other 3 upside down and use them to space the 4 apart. That gives 48 cubes. Then, empty the 4 trays and put 2 ice-cubes in diagonally opposed corners of each of 6 of the trays. Fill the remaining holes — and the entire 7th tray — with water. Using the ice cubes to hold the trays apart, stack all 7 (the 7th tray should go on top), and freeze them. You get another 72 cubes. $\endgroup$ – Amitabh Ghosh Jul 26 '17 at 9:52
  • $\begingroup$ I got one thing wrong, you can get 72 cubes for every batch except the first, for which spacer ice cubes are not yet available. So, that makes 48+72+72+72=264. Again, not sure, if the maths is correct. $\endgroup$ – Amitabh Ghosh Jul 26 '17 at 9:56
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    $\begingroup$ How come it can only handle 7 trays? Stacked vertically that is about 10" high and 6 inches wide, that is truly a tiny, tiny freezer. $\endgroup$ – cybernard Jul 26 '17 at 23:36
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First produce 48 by putting the traces up/down/up/down/up in the freezer. Then use 18 of the ice cubes and put them in the corners of the botton 6 trays, produce 66 more ice cubes. Do this 2 more times and get 246 in total. Possibly could make 264 if you can balance using 2 cubes only.

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  • $\begingroup$ I believe this is the most efficient without modifying the size/shape of any trays. $\endgroup$ – Forklift Jul 26 '17 at 14:49
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    $\begingroup$ @Vikram, this solution is putting 3 ice cubes in 3 ice cube spots in each of the 6 stacked trays below the top tray. Those 3 cubes in each tray act as a triangle base on which the other trays will rest while the other 9 cube slots are filled with water that forms new cubes. $\endgroup$ – Forklift Jul 26 '17 at 16:05
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    $\begingroup$ No, @Vikram, I am assuming the ice cube fits perfectly into the ice cube shaped space in which ice cubes are created. $\endgroup$ – Forklift Jul 26 '17 at 16:32
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    $\begingroup$ @Vikram, Step 1 - do what you said the first 15 minutes. Step 2 - put an ice cube from step 1 in 3 corners of 6 trays, fill the other 9 spots with water and stack them. Fill the 7th tray with water and place on top. Repeat step 2 two more times. $\endgroup$ – Tracy Cramer Jul 26 '17 at 17:57
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    $\begingroup$ Got confirmation. 264 happens to be the correct answer. Thank you! $\endgroup$ – Amitabh Ghosh Jul 31 '17 at 8:14
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Since the freezer can accommodate 7 trays at a time, follow the following arrangement

full tray
upside down empty tray
full tray
upside down empty tray
full tray
upside down empty tray
full tray

So,

in $15$ minutes $4\times 12=48$ cubes can be made and in $60$ minutes $48 \times 4 = 192 $ cubes can be made

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  • $\begingroup$ Hah, I didn't think of that. Nice. $\endgroup$ – Callum Bradbury Jul 26 '17 at 9:07
  • $\begingroup$ I love (hate) "out-of-the-box" answers just as much as everybody else. But when the "mathematics" tag is there, IMO, flipping the tray upside-down doesn't fly. $\endgroup$ – Jeffrey Jul 26 '17 at 17:53
  • $\begingroup$ @Jeffrey,no one cares what you love(hate), if you are so much concerned about tags, why don't you suggest the OP to include logic,lateral-thinking,calculation-puzzle,strategy,optimization tags. do you call adding 4 two digit numbers mathematics? it is arithmetic. $\endgroup$ – user9174 Jul 27 '17 at 4:54
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    $\begingroup$ @vikram he's just upset because he can only do mathematics puzzles, not lateral thought ones, and by tagging this one thus the OP has lured him in under false pretences and wasted 10 seconds of his precious time.. $\endgroup$ – Caius Jard Jul 27 '17 at 7:06
  • $\begingroup$ I don't know should I feel amused or be saddened on which way this discussion is going just coz of tags $\endgroup$ – Amitabh Ghosh Jul 27 '17 at 8:36
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We could freeze more than 192 ice cubes as per @Vikram's answer, IF

We could cut/break the trays.

For example, if the ice cube trays are 3x4, we could cut 3 of the trays to be 3x3:
enter image description here

And then stack the ice trays so that they don't fall into each other:

3x4
3x3
3x4
3x3
3x4
3x3
3x4

Side view:
enter image description here

This gives us $4*4*3 + 3*3*3 = 75$ ice cubes per freezing
And $4*75=300$ ice cubes per hour.

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    $\begingroup$ I suppose if you had an unlimited amount of trays you wouldn't mind breaking some ;) $\endgroup$ – BMS21 Jul 26 '17 at 10:27
  • $\begingroup$ @BMS21 Indeed :)) $\endgroup$ – Gintas K Jul 26 '17 at 10:27
  • $\begingroup$ Standard ice cube trays where I come from are 2x6.... $\endgroup$ – Hellion Jul 26 '17 at 15:10
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    $\begingroup$ @Hellion Sure, but standard ice cube trays where I'm from are also made for stacking so they don't sink into each other ;) $\endgroup$ – Set Big O Jul 26 '17 at 16:45
  • $\begingroup$ @SetBigO a very good point. :-) $\endgroup$ – Hellion Jul 26 '17 at 16:45
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Since we have unlimited supply of trays(and assuming that selling part of them for infinite money and purchasing more freezers is not an option), I would say that in 1 hour we could freeze maximum of

336 ice cubes

ice cubes.

Here's how:

We cut the very top part of X trays and put a non-cut tray into them as long as the height of the cut parts is not exactly equal to the height of the tray. Repeat this for 5 other trays and now you can stack them on each other without the top ones falling through and breaking the bottom ones.


And some pictures:

So we have such an ice tray(okay, it should be 12-cube, but I was too lazy to search for one:
enter image description here

And we will cut the part marked red:
enter image description here
So we will have a lot of these pieces:
enter image description here
Now we only need to cut out enough of these parts to be able to stack trays on top of each other so that trays on top are supported by the trays at the bottom and not on the water/ice itself:
enter image description here

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  • $\begingroup$ The trouble is, these would take up space. $\endgroup$ – wizzwizz4 Jul 26 '17 at 14:11
  • $\begingroup$ @wizzwizz4 no, they wouldn't. Cut it so that the bottom of these pieces per each tray is just as low as the tray's bottom itself, so that maximum length, width and height of each tray remain the same $\endgroup$ – Novarg Jul 26 '17 at 14:13
  • $\begingroup$ I don't understand how that could be done unless you used up space that the ice cubes should occupy. Theoretically, a freezer with space for 7 doesn't even need to have space for "7 and some air around the sides". $\endgroup$ – wizzwizz4 Jul 26 '17 at 14:18
  • $\begingroup$ @wizzwizz4, you just put the cutouts underneath the tray, so that it's bottom surface becomes wider and can support on the top surface of the other tray without falling into it $\endgroup$ – Novarg Jul 26 '17 at 14:30
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    $\begingroup$ Wizzwiz, the suggestion is to cut up X number of trays to make spacers so that the upper tray can sit on top of the lower tray without sinking into it $\endgroup$ – Caius Jard Jul 27 '17 at 7:03
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48

Due to the limitations on placing trays upon partially frozen trays, you can only be freezing 12 cubes at a time. 4*12=48

You can't do 120 cubes at a time, even without the freezing limitation, as it states the freezer can only hold 7 trays, so at best you'd be able to do 84, but that would only be without the placing-upon-unfrozen-trays limitation, which brings it back down to 12 at a time.

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Tilt the freezer 90 degrees so that the door opens up, fill up the entire freezer, like a tub, with water and trays place vertically. The trays should stay separated because they are not falling into each other now. After 15 minutes, remove the trays and crack out the ice cubes, and repeat 3 more times. 7x12x4 = 336 cubes

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  • $\begingroup$ How are you meant to keep the water from sticking multiple trays together? This answer doesn't seem to work. $\endgroup$ – Wen1now Jul 26 '17 at 23:44
  • $\begingroup$ Wonder if PAM cooking spray would work? $\endgroup$ – cybernard Jul 27 '17 at 2:26
  • $\begingroup$ The compressor in a freezer won't work if the freezer is tipped in its side, but I appreciate your lateral thinking $\endgroup$ – Caius Jard Jul 27 '17 at 7:02
  • $\begingroup$ @CaiusJard - could be a solid state Thermoelectric freezer, which should work in almost any orientation. ;) $\endgroup$ – Mark Lakata Jul 27 '17 at 21:55
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Trick question!

Ice cube trays are designed to stack on top of each other without freezing together. There are pieces on the bottoms of the trays that sit on the top lip of the tray below it, so they don't sink in & freeze together.

So you can put all 7 trays in at once.

12 cubes per tray x 7 trays x 4 batches per hour = 336 ice cubes per hour.

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