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Experiment 001: Vertical Math Fest 2023 grade 7 problem 2

(Experiment project is available here.)

In February 2023, yet another Mathematical Festival was held in Moscow and other several cities. It is an mathematical olympiad for 6th and 7th grades that is held each February. There was a problem:

Find any solution of rebus Ф/Е+ВР/АЛЬ=1\text{Ф}/\text{Е} + \text{ВР}/\text{АЛЬ} = 1. Different letters correspond to different digits. Slash means division. As an answer, write down the whole number equality.

(P.S. "ФЕВРАЛЬ" in Russian means "FEBRUARY" when the olympiad was held.) I was asked to find all possible solutions so that the olympiad examiners could check children's works faster.

At first, let's start with idea of the program!​

The idea is simple, we will just go through all possible variants of the letters values. It means that we will go through all 7-permutations (Ф,Е,В,Р,А,Л,Ь)(\text{Ф}, \text{Е}, \text{В}, \text{Р}, \text{А}, \text{Л}, \text{Ь}) of set of all possible digits {0,1,2,3,4,5,6,7,8,9}\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} and check each one of them if it satisfies the rebus.

But there are P(10,7)=10!(10−7)!=604  800P(10, 7) = \frac{10!}{(10-7)!} = 604\;800 possible permutations. We can store them in a list and map/filter that list into a new one and several times to get all sought answers. And there is no problem here because it's not that big number of elements for computer to suffer from. On the contrary, this way is faster than manipulation with sequences for such number of elements. But let us use sequences so that there won't be a problem with future rewriting of the code when there will be much more data to process.

Solution​

First, we create a data class for the permutations and a simple fake constructor from list for it:

data class ФЕВРАЛЬ(val Ф: Int, val Е: Int, val В: Int, val Р: Int, val А: Int, val Л: Int, val Ь: Int)

fun ФЕВРАЛЬ(args: KoneList<Int>): ФЕВРАЛЬ {
    require(args.size == 7u) { "Cannot construct ФЕВРАЛЬ from list $args" }
    val (Ф, Е, В, Р, А, Л, Ь) = args
    return ФЕВРАЛЬ(Ф = Ф, Е = Е, В = В, Р = Р, А = А, Л = Л, Ь = Ь)
}

Then, simple code for getting fractions from the rebus:

val ФЕВРАЛЬ.frac1: Rational get() = Rational(Ф, Е)
val ФЕВРАЛЬ.frac2: Rational get() = Rational(В * 10 + Р, (А * 10 + Л) * 10 + Ь)

Then, let's check that the permutation is a correct:

fun ФЕВРАЛЬ.check(): Boolean = Ф != 0 && Е != 0 && В != 0 && А != 0 && context(Rational.context) { frac1 + frac2 eq one }

Here we check that the first digits are not zeros and check that the sum of the fractions is unit.

Now, let's use the function and get all solutions! At first, let's get all permutations:

val digits = (0 until 10).toKoneList()
val февральPermutations: Sequence<ФЕВРАЛЬ> =
    digits.permutations(7u)
        .map { ФЕВРАЛЬ(it) }

At second, we have just to filter out wrong permutations:

val февральPermutations: Sequence<ФЕВРАЛЬ> =
    digits.permutations(7u)
        .map { ФЕВРАЛЬ(it) }
        .filter { it.check() }

At last, let's sort them by value of the first fraction and then by value of Ф\text{Ф}:

val февральPermutations: Sequence<ФЕВРАЛЬ> =
    digits.permutations(7u)
        .map { ФЕВРАЛЬ(it) }
        .filter { it.check() }
        .sortedWith(
            Comparator<ФЕВРАЛЬ> { perm1, perm2 -> context(Rational.context) { (perm1.frac1 - perm2.frac1).numerator compareTo 0 } }.thenBy { it.Ф }
        )

Now we have all the solutions in февральPermutations variable. Let's print them in 3 different formats:

fun main() {
    println("Ф Е В Р А Л Ь | ФЕВРАЛЬ | Ф/Е+ВР/АЛЬ")
    for (февраль in февральPermutations) println(февраль.run { "$Ф $Е $В $Р $А $Л $Ь | $Ф$Е$В$Р$А$Л$Ь | $Ф/$Е+$В$Р/$А$Л$Ь" })
}

Here are the results:

Ф Е В Р А Л Ь | ФЕВРАЛЬ | Ф/Е+ВР/АЛЬ
2 6 9 0 1 3 5 | 2690135 | 2/6+90/135
2 6 9 8 1 4 7 | 2698147 | 2/6+98/147
3 9 6 8 1 0 2 | 3968102 | 3/9+68/102
3 9 7 2 1 0 8 | 3972108 | 3/9+72/108
3 9 8 4 1 2 6 | 3984126 | 3/9+84/126
3 8 6 5 1 0 4 | 3865104 | 3/8+65/104
3 8 7 5 1 2 0 | 3875120 | 3/8+75/120
2 5 7 8 1 3 0 | 2578130 | 2/5+78/130
4 9 7 0 1 2 6 | 4970126 | 4/9+70/126
2 4 5 3 1 0 6 | 2453106 | 2/4+53/106
2 4 6 5 1 3 0 | 2465130 | 2/4+65/130
2 4 6 9 1 3 8 | 2469138 | 2/4+69/138
2 4 7 8 1 5 6 | 2478156 | 2/4+78/156
2 4 7 9 1 5 8 | 2479158 | 2/4+79/158
2 4 8 5 1 7 0 | 2485170 | 2/4+85/170
2 4 9 3 1 8 6 | 2493186 | 2/4+93/186
3 6 5 2 1 0 4 | 3652104 | 3/6+52/104
3 6 5 4 1 0 8 | 3654108 | 3/6+54/108
3 6 7 9 1 5 8 | 3679158 | 3/6+79/158
3 6 8 5 1 7 0 | 3685170 | 3/6+85/170
3 6 9 2 1 8 4 | 3692184 | 3/6+92/184
4 8 5 3 1 0 6 | 4853106 | 4/8+53/106
4 8 6 5 1 3 0 | 4865130 | 4/8+65/130
4 8 7 6 1 5 2 | 4876152 | 4/8+76/152
3 5 4 8 1 2 0 | 3548120 | 3/5+48/120
3 5 6 8 1 7 0 | 3568170 | 3/5+68/170
3 5 7 2 1 8 0 | 3572180 | 3/5+72/180
3 5 7 6 1 9 0 | 3576190 | 3/5+76/190
3 5 8 4 2 1 0 | 3584210 | 3/5+84/210
3 5 9 6 2 4 0 | 3596240 | 3/5+96/240
5 8 3 9 1 0 4 | 5839104 | 5/8+39/104
2 3 5 8 1 7 4 | 2358174 | 2/3+58/174
4 6 7 3 2 1 9 | 4673219 | 4/6+73/219
4 6 9 1 2 7 3 | 4691273 | 4/6+91/273
6 9 3 4 1 0 2 | 6934102 | 6/9+34/102
6 9 5 8 1 7 4 | 6958174 | 6/9+58/174
6 9 7 8 2 3 4 | 6978234 | 6/9+78/234
6 9 8 1 2 4 3 | 6981243 | 6/9+81/243
5 7 8 6 3 0 1 | 5786301 | 5/7+86/301
3 4 2 7 1 0 8 | 3427108 | 3/4+27/108
6 8 3 5 1 4 0 | 6835140 | 6/8+35/140
6 8 4 3 1 7 2 | 6843172 | 6/8+43/172
6 8 5 1 2 0 4 | 6851204 | 6/8+51/204
7 9 2 4 1 0 8 | 7924108 | 7/9+24/108
7 9 4 8 2 1 6 | 7948216 | 7/9+48/216
7 9 5 8 2 6 1 | 7958261 | 7/9+58/261
4 5 2 6 1 3 0 | 4526130 | 4/5+26/130
4 5 3 2 1 6 0 | 4532160 | 4/5+32/160
4 5 3 6 1 8 0 | 4536180 | 4/5+36/180
4 5 3 8 1 9 0 | 4538190 | 4/5+38/190
4 5 6 2 3 1 0 | 4562310 | 4/5+62/310
4 5 7 2 3 6 0 | 4572360 | 4/5+72/360
4 5 7 6 3 8 0 | 4576380 | 4/5+76/380
4 5 7 8 3 9 0 | 4578390 | 4/5+78/390
5 6 2 9 1 7 4 | 5629174 | 5/6+29/174
6 7 5 9 4 1 3 | 6759413 | 6/7+59/413
7 8 4 5 3 6 0 | 7845360 | 7/8+45/360
7 8 5 2 4 1 6 | 7852416 | 7/8+52/416
7 8 6 3 5 0 4 | 7863504 | 7/8+63/504
7 8 6 4 5 1 2 | 7864512 | 7/8+64/512