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A Bridge Problem (of Sorts)

The website FiveThirtyEight (best known for it's US election predictions) has a weekly article called The Riddler which asks various math and logic problems. One of this weeks problems, The Riddler Classic, is a bridge problem of sorts; asking for the number of legal bridge auctions with and without doubling. If anyone knows this off hand, you can submit it for a chance at internet fame. For those of you who want to calculate it yourself, good luck. If you are only interested in the answer, they will report it on Friday next week.

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