In duplicate bridge there are 204 unique plus scores, ranging from 50 to 7600. Most of the lower scores are common knowledge to every bridge player, while most of the higher ones are rare. Extremely high scores would almost certainly require a bidding accident or outright lunacy — and for the latter affliction, you came to the right web site. Imagine the following scenario:
After opening 2 (strong and artificial) South intends to bid both majors, but East’s jump to 6 puts the screws to that. Under pressure, South takes the optimistic view that a grand slam can be made and cue-bids 7 to force partner to pick a major.
When 7 is doubled and passed around, South redoubles to repeat the demand, only to suffer the unthinkable when lunacy finds its mark. Yes, the defense runs the table, so South is down 13 for the ultimate debacle. Call the paramedics!
South is also to blame, albeit less so than his lunatic partner. The proper course is to pass 6 (forcing after a 2 opening) then bid 6 when partner doubles. Logically this shows both majors, because with only hearts South would bid directly. Six hearts goes down, but it’s a lot better than defending 6 which is cold.
The following table lists every possible duplicate bridge score, except zero (passout). Scores tinted gray are possible only by defeating a contract. Scores tinted gold can be obtained either by making or defeating a contract, hence the remaining majority are possible only by making a contract.
Test your ingenuity by selecting the best answer to each question — or don’t and see if I care.If a question has more than one correct answer, higher scores better.
1. Which bridge score is a Fibonacci number? 0 400 610 750 1120 1440
2. Which bridge score is a Catalan number? 1430 1580 1740 1860 2110 2220
3. Which bridge score is a triangular number? 990 1770 1830 2080 3160 4000
4. Which bridge score is in the first 100 digits of pi? 280 510 640 950 1640 2360
5. Which bridge score has the most divisors? 1680 2160 2560 2880 3120 6400
6. If x and y are bridge scores and 43x = 17y, what is x? 0 170 510 680 1020 1720
Quit
*Contest participants did not have the benefit of multiple choice, which was added for this writeup.
For each problem, the highest (or only) correct answer is awarded 10. Lower but correct answers receive awards of 9 to 5 — credits to Dolly Parton? Problem 5 has only one correct answer, but three others were very close and receive token awards of 7 to 5. The average score of all entries was 52.65, and everyone who scored above average is listed below. Ties are broken by date and time of entry.
In the Fibonacci sequence [0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 …] each successive number is the sum of the preceding two. The only multiple of 10 within range is 610, which is a valid bridge score and the correct answer. However, not the only one: Zero is also a bridge score and receives a token award.
Dan Baker: Other than zero, 610 is the first Fibonacci number to end in zero. Next is 832,040.
Jim Munday: I’d be Fib-bing if I said I have never been minus 610.
Prahalad Rajkumar: This Fibonacci number doubles the fun of bidding and making 4 NT (nonvul).
Manoj Kumar K: Every 15th Fibonacci number ends in zero.
Really? Let’s see… F0 = 0; F15 = 610; F30 = 832,040; F45 = 1,134,903,170; F60 = 1,548,008,755,920… Never doubt an Indian about math!
In the Catalan sequence [1 1 2 5 14 42 132 429 1430 4862 16796 …] the formula for successive numbers is more complicated — if you’re interested in the details, please enroll in my upcoming course RP Math 101. Regardless, it is easily seen that 1430 is the only bridge score. As an added bonus, Kantar-style responses* to RKC can now be described as “Catalan.”
*Eddie Kantar was a great player/writer, but his “1430” flip-flop was a disastrous addition to bidding theory. The simple task of counting from zero to one is now shrouded by which method is being played, causing endless misunderstandings, even among experts.
Dan Baker: The first Catalan number to end in zero is 1430; next is 742,900.
Prahalad Rajkumar: Not a Catalan opening in Chess, but definitely an RKC ask in bridge!
Jonathan Mestel: Mille quatre-cent trente.
In the triangular sequence [1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 …] the gap between numbers increases by one each time. I’ll spare you the complete list within range, but 12 triangular numbers coincide with valid bridge scores as shown below.
Dan Baker: T(n) = n(n+1)/2 ends in zero for n = 0, 4, 15 or 19 mod 20. There are 12 triangular bridge scores, and the highest is T(79) = 79×80/2 = 3160.
Venk Natarajan: Since most multiples of 10 are bridge scores until you get past small slams, they are triangular when the [expression] n(n+1)/2 has enough multiples of 2 and 5 in n and n+1.
Now he shows up. Where was Venk for my Venusian Victory?
Nineteen bridge scores (colored red) appear in the first 100 digits* of pi, but eight of these are the digit zero by itself (passout) which is hardly worth an award when the goal was to find the highest. The 11 valid plus scores are ranked in the table.
*Custom is to count digits after the decimal point, but it wouldn’t matter here if you counted the initial 3.
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679
Dan Baker: There are eight zeros in the first 100 digits, and only one is preceded by a valid four-digit score (digits 68-71). Next largest is 950 (digits 30-32).
Venk Natarajan: It takes surprisingly long for the first zero to show up in pi.
Prahalad Rajkumar: Bridge is the true Life of Pi, redoubled. Let the tiger loose, and ride the waves to victory.
In mathematics, the divisors (or factors) of a number are those leaving no remainder after division, which include 1 and the number itself. For example, 100 has nine divisors [1 2 4 5 10 20 25 50 100]. Finding the bridge score with the most divisors may have taken some effort, but there is a clear winner. Three other scores are close runners-up and receive token awards.
Counting the number of divisors can be tedious, but there is a formula to determine this from a number’s prime factors.* Increase the exponent of each prime factor by one, then multiply the resultants. For example, 2880 = 26 × 32 × 5, hence the exponents are [6 2 1]. Increasing each by one gives 7 × 3 × 2 = 42 divisors. Curiously, the formula even works for the number 1, whose “prime factorization” could be shown as any prime (or primes) to the zero power, e.g., 1 = 20, so incrementing the exponent gives one divisor.
*Note that prime factors do not include 1, because 1 is not a prime number.
Dan Baker: Counting 1 and itself, 2880 has 42 divisors (2520 would have it beat at 48 if it were a possible score); 3120, 2160, and 1680 are next with 40 divisors; 50 has the fewest with only six.
David Wu: This was a tough problem; feels like 2880 should be the answer, but how to be sure? Fortunately, 1680 is highly composite, so I only had to check scores upward from there.
Richard Stein: Factors 26 × 32 × 5 produce 42 divisors.
Prahalad Rajkumar: Divisors multiply like your opponent’s vulnerable overtricks in 2 NT redoubled making seven!
For this equality to exist, score x must be divisible by 17 (and 170 because all scores are multiples of 10) and score y must be divisible by 43 (and 430). Since 170 and 430 are both valid bridge scores, the simplest solution is 43 × 170 = 17 × 430, so x = 170. Multiples of these numbers reveal the two superior solutions below, and of course, zero for both x and y cannot be denied as correct.
Dan Baker: Only three bridge scores are divisible by 43 and therefore possible values of y: 430, 1290, and 1720. All three correspond to valid scores for x (170, 510, 680).
Jim Munday: Best solution is 43 × 680 = 17 × 1720 = 29,240.
Charles Blair: It took me several weeks to realize that ‘43x’ was not required to be a bridge score.
Ah! Now I understand why his first entry had the only solution being zeros.
For dessert, I will offer some crucial information to boost your bridge scores:
In the first 4 GB (4,294,967,296) digits of pi: There are 429,497,778 zeros and 566,934,531 bridge scores (not counting zero alone). The first score to occur is 950 (digit 30) and the one taking longest to occur is 2320 (digit 49521). The most common nonzero score is 90, occurring 42,955,219 times. Of course, any two-digit sequence occurs about 10 times more often than three digits, or 100 times more often than four. The most common three-digit score is 180 (4,299,089 times). Remarkably, the most common four-digit score is 7600 (432,026 times) which I submit as mathematical proof that pi is not transcendental but just a disaster waiting to happen. Welcome to the Club.
Now if you think my efforts are useless, get a load of these guys:
Charles Blair: When the last Great Scorer cometh… Hi-ho, hi-ho, it’s off to work we go!
Prahalad Rajkumar: In math we must follow rules, but in bridge we just break them and rewrite them.
Jim Munday: Is there a bonus for achieving these scores?
© 2025 Richard Pavlicek
by Richard Pavlicek by Richard Pavlicek