Lev Landau - 2 rubles
Landau's license plate game
Home of its first PC implementation - Dau: the game

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Definition taken from M. I. Kaganov: "The point of the game is to make an equality out of any four-digit license plate number (Soviet license plate numbers are of the form "AB-CD" - for example, "12-34".) Here are the rules: you can use only the arithmetic, algebraic, and trigonometric operations learned in school, you're not allowed to rearrange the numbers, and you have to work out the solution in your head. In other words, you have to turn "-" into "=" by inserting signs known to any high school student [plus, minus, times, divides, square root, logarithm, sine, cosine, and so on] between the numbers." Examples: 75-31 (7-5=3-1), 75-33 (7-5)=3!/3. Of course, there can be other solutions, as you can easily see.

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Question of allowed functions in Landau's game is a tough one - differentiation, whole and fractional parts are not allowed for instance - differentiation is dismissed as non-elementary, while whole and fractional parts are dismissed as not a part of standard high school curriculum. You may notice that they would actually trivialize the game completely - so it's better they are out. As the curriculum changes, the allowed set of functions and operations changes too.

It is natural to ask - is there a panacea: a formula that could solve any license plate? Landau's first answer was negative - he assumed so since he wasn't able to solve all license plates. One would say that if there is such a solution, it would ruin the game, while others would find a new charm in the game - pursuing other general solutions, taking in some complex mathematics and playing it in their own way.

So, let's ask once again - are there general solutions, for all license plates?

Spoiler alert!

As a matter of fact, there are such general solutions. If you want to work it out for yourself, stop reading right here - and go play Dau: the game. If you still want to know what it is, you'll find below 3 general solutions for the game.

Just scroll down to find out what the general solutions are (it's not too late to stop scrolling if you don't like spoilers, though!)

General solutions (proofs can be found in articles listed in Online resources section)

1.a. Y. Gandel's trigonometry: It's easy to prove that this equation holds

eq 1

Using it, it's easy to turn any square root of any non-negative integer into square root of its successor. Iterated use makes it possible to turn any such number into any other - and therefore turn the square root of left license plate side in the square root of right license plate side.

1.b. S. Fomin's trigonometry:  As B. Gorobec noted, secant is not in the official Russian high school curriculum, so Landau's game purists would reject this solution - but S. Fomin salvaged it, writing it in this form:

Eq 2

2. B. Gorobec's trigonometry with factorials: Since 6!=720=2x360, and we know that sin(360k) for integer k equals zero, it's enough to make a number greater than 6 on both sides of the license plate and apply the sine of factorials on both sides, noting that we use sine with degrees as arguments.

3. My own binary logarithm: Since for almost all pairs of digits which can appear on one side of the license plate have either sum, difference or quotient equal to 0, 1, 2 or 4 (only 5 pairs don't have that property, but they're easily reduced to zero using square root and/or factorial), they can be reduced to zero using logarithm with base 2. You might ask how can we use log with base 2, since that would mean us using an added digit '2' which isn't allowed? Well, according to ISO standard, binary algorithm is in that special group with algorithms with base 10 and e - and they have their special symbols - ln for base e, lg for base 10, and lb for base 2. Another remark may be - is lb taught in schools? Well, if it isn't, it should be - knowing how important lb is in information technology, and how important information technology is in our lives in 21st century!