the cross was first used in christianity in from the 4th century.
the cross was first used in christianity in from the 4th century.
when was the cross first used in christianity?
from the 4th century
Na
Na
assume that a solution minimizes the total number of crossings. this gives a total of five crossings - three pair crossings and two solo-crossings. also, assume we always choose the fastest for the solo-cross. first, we show that if the two slowest persons (c and d) cross separately, they accumulate a total crossing time of 15. this is done by taking persons a, c, & d: c+a+d+a = 5+1+8+1=15. (here we use a because we know that using a to cross both c and d separately is the most efficient.) but, the time has elapsed and person a and b are still on the starting side of the bridge and must cross. so it is not possible for the two slowest (c & d) to cross separately. second, we show that in order for c and d to cross together that they need to cross on the second pair-cross: i.e. not c or d, so a and b, must cross together first. remember our assumption at the beginning states that we should minimize crossings and so we have five crossings - 3 pair-crossings and 2 single crossings. assume that c and d cross first. but then c or d must cross back to bring the torch to the other side, and so whoever solo-crossed must cross again. hence, they will cross separately. also, it is impossible for them to cross together last, since this implies that one of them must have crossed previously, otherwise there would be three persons total on the start side. so, since there are only three choices for the pair-crossings and c and d cannot cross first or last, they must cross together on the second, or middle, pair-crossing. putting all this together, a and b must cross first, since we know c and d cannot and we are minimizing crossings. then, a must cross next, since we assume we should choose the fastest to make the solo-cross. then we are at the second, or middle, pair-crossing so c and d must go. then we choose to send the fastest back, which is b. a and b are now on the start side and must cross for the last pair-crossing. this gives us, b+a+d+b+b = 2+1+8+2+2 = 15.
bridge and torch problem