Arithmetic Formulas

Arithmetic Formulas

By Stephen Meskin

As actuaries, we are all pretty good at plugging numbers into a formula and getting a result. In this set of problems, I am asking you to go the other way: Given a specific input (three digits), you are to find a formula that uses those digits to produce a prespecified output. Sometimes more than one formula will work, but you need not find them all.

Sound easy? To make it interesting, there are constraints on what can be used to construct a formula. Even though the answers are short, they may be hard to come up with, so I have provided 10 answers for you to study, a variety of at least 22 problems to pick from, and you will have done a good job if you get 11 of them, so that will count as solving this problem set.

To begin with, each of the three input digits must be used once and only once in the formula. On the other hand, the arithmetic operations: sum (+), difference and negation (-), multiplication (*), division (/), exponentiation (^), decimal points(.), parentheses ((,)), and concatenation (adjoining two digits; for example, taking 1 and 2 to make 21 or 1.2) may be used finitely often. Decimal points and concatenation can only be applied to digits, all the other operations can be applied to expressions as well. Within these constraints, you can try the problems in Section I.

The * after F indicates that I’m aware of more than one solution to item F. Distinguishing solutions is a matter of judgment. For example, some might say that (3-.1)/.1 is a second one for item G, but in my opinion, it is essentially the same as the one given. See, for example, item B of Section II.

In Section II, three more operations can be used:

  • Factorials: Factorials should only be applied to integers, that can include expressions equal to an integer; 0!=1; the factorials of 1 and 2 add nothing so they shouldn’t be used. The symbol !! is interpreted to mean the factorial of a factorial, i.e., 3!!=6! (this differs from standard practice in number theory).
  • Roots: Using the root symbol (√) without an index means square root, but no 2 is needed from the input set; otherwise, the index can be any expression. When using 3 √, a 3 is needed from the input set.
  • Repeating decimals: In repeating decimals, a line is drawn under digits to show that they repeat endlessly. For example, .7=.777… Repeating decimals can only be applied to digits.

Solutions may be emailed to puzzles@actuary.org. In order to make the solver list, your solutions must be received by June 1, 2024.

Answers to Previous Puzzles: The Not-So-Friendly Confines

1. What is the maximum number of double plays that can occur in a baseball game that does not go to extra innings? With only one possible in a half inning, the answer is 9 × 2 = 18.

2. What is the maximum number of hits there can be in an inning if no runs get scored? If the first two runners get a single and get caught stealing second base, the next three batters can get a single. So we are up to five hits with two outs. If the next batter lines a ball that hits a baserunner, the batter gets awarded a hit but the on-base runner is out and no runs are scored. If this sequence happens in both halves of the inning, there could be 12 hits without a run.

3. What is the minimum number of pitches required to complete an inning of baseball? This answer changed a few years ago due to new rules. There are at least two ways for the answer to be zero. One is to have the batter intentionally walked and then picked off. The second is for a pitcher to have four time-clock violations, resulting in a walk, followed by the batter getting picked off.

4. What is the maximum number of pitchers that can be used to complete a combined, 9-inning no-hitter? On Sept. 1, rosters expand to 14 hitters and 14 pitchers. While there is a three-batter rule in place, if the first pitcher walked three batters and the next pitcher threw three strikeouts, you can have more than one pitcher complete an inning. And because runs can score in a no-hitter, you can in theory use every person on the roster. So, this leads to a max of 14 players if only pitchers pitch, and 28 if batters could pitch as well. The answer for a combined perfect game would be different.

5. Assuming there are no hits, errors, walks, balks, passed balls or wild pitches, what is the maximum number of runs that can be scored in an inning? As I didn’t preclude hit-by-pitches in the question, in theory the maximum is infinite. If HBPs were disallowed as well, the answer would be two, as it’s possible to score the ghost runner in both halves of the extra inning. Note that catcher interference is an error, so you can’t use that loophole to get infinite runs.


Andy Boyer, Bob Conger, Rui Gio, Tim Grusenmeyer, George Harrison, Steve Itelson, Clive Keating, Paul Lasky, Rob McCleish, Jim Muza, Albert Perez, Glen Reineke, Adam Reiner, Anthony Salis, Noam Segal Al Spooner, David Tate, Daniel Wade, Brian Zange, Steve Zeske, and the family of Bill, Bob, and Jim Feldman.

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