Kvant for Younger School-Children
Kvant is a very popular Russian math and physics journal for high school-children. My favorite page is the one with puzzles directed to younger readers. Here are two puzzles from the latest online issue: 2012 number 3.
The first one, by N. Netrusova, is optimistic about the next year.
An astrologist believes that a year is happy if its digit representation contains four consecutive digits. For example, the next year, 2013, will be happy. When was the previous happy year?
The second problem is by L. Mednikov and A. Shapovalov. It confused me at first. For a moment I thought that the best answer is 241 rubles:
A big candle lasts one hour and costs 60 rubles. A small candle lasts 11 minutes and costs 11 rubles. Can you measure a minute by spending not more than a) 200 rubles, b) 150 rubles?
Saying “matches are free” could serve as a hint.29 July 2012, 1:32 pm
1432, for first….but for second I am still thinking 241!30 July 2012, 5:53 am
Can I burn both the ends of the candle…to reduce time in half? Seems similar to a rope puzzle…30 July 2012, 5:56 am
148 for the second30 July 2012, 7:12 am
BTW, it seems to me that once you can do better than 241 you can do better than 150. What is the point in a?30 July 2012, 7:22 am
Ilya, my first attempt was 186 rubles. Then, I figured out the solution for 148 rubles.30 July 2012, 11:40 am
I give up! How do you do better than 241 rubles?
Yes I did assume matches are free, so that you can snuff out candles and resume burning them from where you left off, but still I don’t get it.
(I hope it’s not something stupid like burning candles at both ends. Candles are not ropes, and most candles I’ve seen can only be burnt at one end, at least if you want their “duration” to be reliable.)30 July 2012, 11:47 am
a) 1432. This astrologer hasn’t had many happy years recently.
b) 126 rubles.
Light the hour candle and an eleven minute candle. Continue re-lighting 11 minute candles until the hour candle only has five minutes left. Extinguish the hour candle. Take an additional 11 minute candle and cut it so that you have a five minute and a six minute portion (measuring against the five minutes left on the hour candle. Light either five minute candle and the six minute candle. Start the timer when the five minute candle burns out.30 July 2012, 3:38 pm
I think that the assumption is that candles do not furn proportionally. They might burn randomly. Otherwise, I love your solution. BTW it requires one more 11-minute candle, so the answer would be 137.30 July 2012, 6:59 pm
No, Brenda’s solution requires 126 rubles only: one 60-minute candle, five 11-minute candles, and then another 11-minute candle. This last candle is split into a 5-minute candle and a 6-minute candle, and then both are lit simultaneously. So the total cost in rubles is (1)(60) + (5+1)(11) = 126.31 July 2012, 12:18 am
But yes, the assumption that we can extract a 5-minute candle out of the 11-minute candle by looking at the 5-minute remnant of the 60-minute candle is a bit suspect (e.g. the 60-minute candle might be thicker, so lengths are not comparable), though I find it more palatable than burning a candle at both ends. :p
Unforunately I was lazy enough to stop at 186 rubels… and I checked the comments. Of course, when you know that there is a better solution (and what it is), finding it is very simple. I was mad at myself for not thinking longer… Nice problems though!31 July 2012, 6:12 am
Now I understand Brenda’s solution.31 July 2012, 8:27 am
If the problem had allowed you to divide an eleven-minute candle into a five-minute piece and a six-minute piece, one minute could have been measured for 11 rubles.1 August 2012, 12:52 am
Bill: The trick in Brenda’s nice solution is that you first obtain a five-minute piece out of the 60-minute candle, and *then* use it to extract the corresponding five-minute piece from the 11-minute candle (by cutting it at the same length, assuming that both candles have the same thickness). It’s not the same as cutting out a five-minute piece out of thin air, based on just guesswork. The candles don’t even have to burn proportional to the length; they only need to have the property that two candles of the same length take the same time to burn, and that splitting a candle into two parts preserves the total duration.
But this solution suffers from the same problem as burning a candle at both ends: candles usually have their wick at only one end; if you want to burn one from the other end you may have to whittle out quite a bit of wax to get to the wick, and the duration is no longer reliable. Worse, candles come with weird shapes and extra thickness at the bottom. It is for these reasons that I still hope there is some solution that does not involve burning at the other end… I have not been able to think of anything better than 241. 🙁1 August 2012, 12:59 am
ano: Light the hour candle and an eleven minute candle. Continue re-lighting 11 minute candles until the hour candle only has five minutes left. Light 2 eleven minute candles. Extinguish them when the hour candle is burned off. Now you have 2 six minute candles. Light an eleven minutes candle and one of the six minutes candles, and then the other one. All together is 60+8*11=148.1 August 2012, 1:50 am
1. burning at both ends allowed:
– 104 rubles: light 60 min candle at both ends, for 22 minutes using 2 11 minute candles lighted at only one end. After this 16 minutes remain from the big candle. light both the big candle and a small candle at only one end, and let burn until the small one ends (11 min). Now 5 minutes remain from the big candle, light the big candle from one end and a small candle from both ends, and when the big candle is burned completely you will have one minute left in the small candle. Total cost: 1 big candle, 4 small ones = 104 rubles.
2. burning at both ends not allowed, and no cutting of the candles.1 August 2012, 4:35 am
similar to Brenda’s solution. Burn the big candle and 5 small candles until you remain with 5 minutes on the big one. Light two small candles until the big one burns out. You will have two small candles each with 6 minute remaining. Use another small candle with one of the 6 minutes candle to get 5 minutes remaining on a small candle. And now you have 2 small candles one with 5 minutes left and one with 6 minutes left. Lighting them at the same time would yield one minute. Total cost: 1 big candle, 8 small ones = 148 rubles
Ilya: That’s very nice, thanks! Now I feel stupid for artificially restricting myself to having only two candles lit at a time, and/or always using a new candle each time.1 August 2012, 4:39 am
Gotcha. I should have read more carefully.
For completeness, my original 186-ruble solution (for question a):
Light two one-hour candles and one eleven-minute candle. Sequentially light eleven-minute candles until the one-hour candles have only five minutes left. Extinguish one of them and light another eleven-minute candle. When the first one-hour candle burns out, re-light the other one-hour candle. When it burns out, the last eleven-minute candle will have one minute left to burn.1 August 2012, 7:34 am
This is great! I’m giving the first one to my son (he’s just out of 1st grade). I will report progress later on.7 August 2012, 1:58 am
Why isn’t the first answer 1908?13 August 2012, 5:51 pm
Or 1920?13 August 2012, 10:08 pm
or 1980?9 November 2012, 3:35 pm
The digits shouldn’t go around.9 November 2012, 6:17 pm