New Year’s Day
Now is the accepted time to make your regular annual good resolutions. Next week you can begin paving hell with them as usual.
Yesterday, everybody smoked his last cigar, took his last drink, and swore his last oath. Today, we are a pious and exemplary community. Thirty days from now, we shall have cast our reformation to the winds and gone to cutting our ancient shortcomings considerably shorter than ever. We shall also reflect pleasantly upon how we did the same old thing last year about this time.
However, go in, community. New Year’s is a harmless annual institution, of no particular use to anybody save as a scapegoat for promiscuous drunks, and friendly calls, and humbug resolutions, and we wish you to enjoy it with a looseness suited to the greatness of the occasion.
For many homeschoolers, January is the time to assess our progress and make a few New Semester’s Resolutions. This year, we resolve to challenge ourselves to more math puzzles. Would you like to join us? Pump up your mental muscles with the 2013 Mathematics Game!
Rules of the Game
Use the digits in the year 2013 to write mathematical expressions for the counting numbers 1 through 100.
- You must use all four digits. You may not use any other numbers.
- You may use +, -, x, ÷, sqrt (square root), ^ (raise to a power), ! (factorial), and parentheses, brackets, or other grouping symbols.
- You may use a decimal point to create numbers such as .2, .01, etc.
- You may create multi-digit numbers such as 10 or 203, but we prefer solutions that avoid them.
You may use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal.
You may use multifactorials:
- n!! = a double factorial = the product of all integers from 1 to n that have the same parity (odd or even) as n.
- n!!! = a triple factorial = the product of all integers from 1 to n that are equal to n mod 3
[Note to teachers: Math Forum modified their rules to allow double factorials, but as far as I know, they do not allow repeating decimals or triple factorials.]
How To Play
With only small numbers to work with this year, we will need every trick in the book to create variety in our numbers. Experiment with decimals, double-digit numbers, and factorials of all sorts. Remember that dividing (or using a negative exponent) creates the reciprocal of a fraction, which can flip the denominator up where it may be more helpful.
Use the comments section below to share the numbers you find, but don’t spoil the game by telling us how you made them. You may give relatively cryptic hints, especially for the more difficult numbers, but be warned: Many teachers use this puzzle as a classroom assignment, and there will always be students looking for people to do their homework for them.
- Do not post your solutions. I will delete them.
There is no authoritative answer key for the year game, so we will rely on our collective wisdom to decide when we’re done. We’ve had some lively discussions the last few years. I’m looking forward to this year’s fun!
As players report their game results below, I will keep a running tally of confirmed results (numbers found by two or more players). Today is Kitten’s birthday, however, so I won’t spend much time at my computer. Also, I’ll be traveling a lot this month, so this tally will lag a few days behind the results posted in the comments.
Percent confirmed = 100%!
[The number 83
is under reviewhas been confirmed.]
Reported but not confirmed = 83.
Numbers we are still missing = none.
And if you would like to join me in the “extended edition” game…
The following data are incomplete, pending further investigation:
Middle school rules = 92%.
1–42, 44–75, 77-81, 84, 89–91, 92–100.
Old Math Forum rules, no repeating decimals or multifactorials.
New Math Forum rules = 98%.
All the above, plus 43, 76, 82, 85-87.
NOT Math Forum:
Needed multi-digit numbers: none?
[I’m still relying on multi-digits for several numbers, but Bill in the comments says he found ways around them.]
Could NOT keep the digits in order:
34, 37-38, 41, 45, 49, 51-52, 54-55, 59, 61, 65, 68, 76, 79, 81-84, 86-91, 93-95, 97-99.
Math Forum will begin publishing student solutions after February 1, 2013. Remember, your students may not submit answers with triple (or higher) factorials or repeating decimals to the Math Forum site.
Clarifying the Do’s and Don’ts
Finally, here are a few rules that players have found confusing in past years.
These things ARE allowed:
- 0! = 1. [See Dr. Math’s Why does 0 factorial equal 1?]
- The only digits that you can use to build 2-or-more-digit numerals or decimals are the standard base-10 digits 2, 0, 1, 3.
- Unary negatives count. That is, you may use a “-” sign to create a negative number.
- You may use (n!)!, a nested factorial — a factorial of a factorial. Nested square roots are also allowed.
- The multifactorial n!k = the product of all integers from 1 to n that are equal to n mod k. You may write the double factorial and triple factorial as !! and !!!, respectively, but for higher multifactorials BOTH n and k must be constructed from the year digits.
These things are NOT allowed:
- “0!” is not a digit, so it cannot be used to create a base-10 numeral.
- The decimal point is not an operation that can be applied to other mathematical expressions: “.(0!)” does not make sense.
- You may not use any exponent unless you create it from the digits 2, 0, 1, 3. You may not use a square function, but you may use “^2”. You may not use a cube function, but you may use “^(2+1)”. You may not use a reciprocal function, but you may use “^(-1)”.
- You have to “hit” each number from 1 to 100 exactly, without rounding off or truncating decimals. You may not use the integer, floor, or ceiling functions.
- 2013 Mathematics Game Worksheet
For keeping track of which numbers you’ve solved.
- 2013 Mathematics Game Manipulatives
This may help visual or hands-on thinkers.
- 2013 Mathematics Game Student Submissions
For elementary through high school students who wish to share their solutions.
For more tips, check out this comment from the 2008 game.
Heiner Marxen has compiled hints and results for past years (and for the related Four 4’s puzzle). Dave Rusin describes a related card game, Krypto, which is much like my Target Number game. And Alexander Bogomolny offers a great collection of similar puzzles on his Make An Identity page.