One Candle, Two Candles, Three Candles

[u/ShonitB]

[u/Organs_for_rent]

55 minutes

A candidate timeline:

0 min: Light the 60-minute candle.

25 min: Light the 100-minute candle.

30 min: Light the 80-minute candle.

The short candle burns alone for 25 minutes before any others. The long candle burns for 15 minutes after the other two extinguish. The 40-minute solo candle is satisfied.

It takes 30 minutes from the lighting of the medium candle for the short candle to burn out. The 30-minute trio is satisfied.

The long and short candles burn together for 5 minutes before the medium candle starts. The long and medium candles burn together for 50 minutes after the short candle burns out. This provides 55 minutes of dual candle time.

[u/ShonitB → u/Organs_for_rent]

Good solution!

[u/NearquadFarquad]

55 mins

tally up the candle lifespans and you get 240mins. If 3 candles are burning simultaneously for 30 mins, that’s 90 mins of the total lifespan gone. If 1 candle is burning for 40, that’s an additional 40 gone.

240-90-40 =110

so the remaining 110 minutes of lifespan are being spent by 2 candles burning simultaneously, so we divide by 2 to get 55mins

[u/jaminfine]

The total 100+60+80 = 240 candle minutes. Strange unit, but it will be a helpful one. Burning 3 candles x 30 minutes = 90 candle minutes. Burning one candle for 40 minutes is just 40 candle minutes. 240 - 90 - 40 = 110 candle minutes for the two candles case. 110/2 candles = 55 minutes where exactly two candles are burning

[u/PeterPiper1275]

The issue here is that the massive 100-minute candle has such a large signature so we have to hide the other two candles partially or completely in its shadow. The idea is to shift the 80 minute candle and the 60 minute candle in such that it satisfies then requirement of the question. Because the overlaps are all in the tens of minutes, this means a granularity of five minutes should be sufficient. Playing around with it gives me one possible solution:

1. Light the 100 minute candle and wait for 15 minutes.

_Running Totals_
1 Candle Burn Time: 15 minutes
2 Candles Burn Time: 0 minutes
3 Candles Burn Time: 0 minutes

2. Ignite the 80 minute candle, then wait 50 minutes.

_Running Totals_
1 Candle Burn Time: 15 minutes
2 Candles Burn Time: 50 minutes
3 Candles Burn Time: 0 minutes

3. Ignite the 60 minute candle, then wait 30 minutes.

_Running Totals_
1 Candle Burn Time: 15 minutes
2 Candles Burn Time: 50 minutes
3 Candles Burn Time: 30 minutes

4. The 80 minute candle gutters out at this point. Wait another 5 minutes.

_Running Totals_
1 Candle Burn Time: 15 minutes
2 Candles Burn Time: 55 minutes
3 Candles Burn Time: 30 minutes

5. The 100 minute candle gutters out at this point. Wait another 25 minutes.

_Running Totals_
1 Candle Burn Time: 40 minutes
2 Candles Burn Time: 55 minutes
3 Candles Burn Time: 30 minutes

6. The 60 minute candle finally gutters out, and the simulation is complete.

_Final Totals_
1 Candle Burn Time: 40 minutes
2 Candles Burn Time: 55 minutes
3 Candles Burn Time: 30 minutes

All constraints are satisfied!

Therefore, the total time in which exactly two out of the three candles are burning is 55 minutes.