Five Bridges and a Storm — A Perplexing Probability Puzzle
Wednesday 14 January 2026 at 23:21
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Edited by Richard Walker, Thursday 15 January 2026 at 23:18
Here's a probability question I found in the marvellous problem collection Cut the Knot.
In Figure 1 (a) we see two islands in a river and five bridges joining the islands to each bank and to one another.
Unfortunately a recent storm has caused damaged and each bridge has a 0.5 probability of having been swept away. What is the probability that a traveller can still cross the river?
This was harder than I was expecting and I found it quite confusing to think about. It did remind me though of a celebrated problem, the seven bridges of Königsberg, which prompted to dispense with some of the details and draw Fig. 1 (b), which captures the structure, and then after a bit of head-scratch I reasoned as follows.
Each bridge is either intact or has been swept away, with an equal 0.5 probability of each. So there are 2x2x2x2x2 = 32 equally likely possibilities, not too many. So I could systematically enumerate the ones that allowed a crossing, which I did, and then just count them and divide by 32. Here they are (Figure 2), where the thick lines represent bridges that have survived the storm. I hope I have included them all!
So the probability is . I looked up the answer and sure enough 0.5 is the probability. But the problem setter had a much cleverer way of finding the answer!
Suppose a steamboat is coming along the river.
The dotted lines show the possible channels the boat might follow, if the corresponding bridge is has been swept away. Moreover if we draw a diagram of these channels and their interconnection it turns to be structurally identical to Figure 1 (b), and the boat will be able to pass along each channel with probability 0.5, because if the probability a bridge is intact is 0.5, the probability it has been swept away is also 0.5.
So probability(boat can pass along the river) = probability(a traveller can cross over the river)
And now for the masterstroke! If the boat can pass there must be an unbroken path along the river and the traveller is cut off. If the traveller can cross there must be an unbroken path across the river and the boat is cut off. So the two probabilities above must add up to 1 and hence both are 0.5.
This is a very pretty solution. It seems to have originated in a book Bas [1], as an example where arguing from symmetry gives a correct probability, which in some cases it may not.
And never forget the old malaprop: Don't burn your bridges before they're hatched.
[1] Bas C. van Fraassen, Laws and Symmetry, Oxford University Press, 1989
Five Bridges and a Storm — A Perplexing Probability Puzzle
Here's a probability question I found in the marvellous problem collection Cut the Knot.
In Figure 1 (a) we see two islands in a river and five bridges joining the islands to each bank and to one another.
Unfortunately a recent storm has caused damaged and each bridge has a 0.5 probability of having been swept away. What is the probability that a traveller can still cross the river?
This was harder than I was expecting and I found it quite confusing to think about. It did remind me though of a celebrated problem, the seven bridges of Königsberg, which prompted to dispense with some of the details and draw Fig. 1 (b), which captures the structure, and then after a bit of head-scratch I reasoned as follows.
Each bridge is either intact or has been swept away, with an equal 0.5 probability of each. So there are 2x2x2x2x2 = 32 equally likely possibilities, not too many. So I could systematically enumerate the ones that allowed a crossing, which I did, and then just count them and divide by 32. Here they are (Figure 2), where the thick lines represent bridges that have survived the storm. I hope I have included them all!
So the probability is . I looked up the answer and sure enough 0.5 is the probability. But the problem setter had a much cleverer way of finding the answer!
Suppose a steamboat is coming along the river.
The dotted lines show the possible channels the boat might follow, if the corresponding bridge is has been swept away. Moreover if we draw a diagram of these channels and their interconnection it turns to be structurally identical to Figure 1 (b), and the boat will be able to pass along each channel with probability 0.5, because if the probability a bridge is intact is 0.5, the probability it has been swept away is also 0.5.
So probability(boat can pass along the river) = probability(a traveller can cross over the river)
And now for the masterstroke! If the boat can pass there must be an unbroken path along the river and the traveller is cut off. If the traveller can cross there must be an unbroken path across the river and the boat is cut off. So the two probabilities above must add up to 1 and hence both are 0.5.
This is a very pretty solution. It seems to have originated in a book Bas [1], as an example where arguing from symmetry gives a correct probability, which in some cases it may not.
And never forget the old malaprop: Don't burn your bridges before they're hatched.
[1] Bas C. van Fraassen, Laws and Symmetry, Oxford University Press, 1989