Czech federation of mancala games

Competition, part 1

Toguz kumalak/Toguz korgool, is a two-player game. We use a board that has two rows of nine cups/holes, and nine seeds in each cup at the beginning. (Toguz = nine, kumalak/korgool = seed, ball.) We will show the basics of this game on its smaller variant with five cups on each side and five seeds in each cup.

Here is the small board with the initial position. Two rows with five cups, five seeds in each cup, and in the middle so far empty treasury cups.

T5T4T3T2T1


B1B2B3B4B5

You are sitting at the bottom side of board, so the lower side of the cups is yours. And it is your turn to play.

For better understanding we show what will happen using position with only one full cup to sow. So we sow the cup B3.

T5T4T3T2T1


B1B2B3B4B5
T5T4T3T2T1


B1B2B3B4B5

What happened here? We chose the cup B3, we left one seed in it, and we put one seed into each cup while going around the board in anticlockwise direction. So the first seed went into cup B3, the second to B4, third to B5, fourth to T1 and fifth to T2.

Control question: If we sow the cup B2, how will the position look like after sowing? Will it look like as at the picture A or B?

Initial position A B
T5T4T3T2T1 T5T4T3T2T1 T5T4T3T2T1



B1B2B3B4B5 B1B2B3B4B5 B1B2B3B4B5

Correct answer is B.
At the picture A the bottom player did not sow the seeds according to the rules, he skipped some cups and he put more than one seed into some cups. All this is forbiden.
At the picture B the bottom player sow correctly, one cup after another, just one seed in each, and did not skip any cup.

Once more from the initial position: We sow the cup B3. We will go around the board and we put one seed into each cup.

T5T4T3T2T1


B1B2B3B4B5
T5T4T3T2T1


B1B2B3B4B5

After sowing the position will look like on the picture.

We showed the rule for sowing and now we show how to count, where we put last seed.

First method:
Add up number of seeds in the cup you want to sow and it's number and then deduct one. After that, you have to deduct 5 so many times, so that you get number between 1 and 5. After these operations you get number of the cup, where the sowing ends.
Practical example:
We want to sow cup B4, which contains 12 seeds.
We count:
12 seed + 4 (cup number 4) - 1.
12 + 4 - 1 = 15.
It is more than 5, so we deduct 5.
15 - 5 = 10.
It is still more than 5, so we deduct 5 again.
10 - 5 = 5.
Now we have number between 1 and 5, concretly 5. So the sowing will end up in cup 5.

Second method:
We take a look to number of the opposite cup, or we count this number as number of cups in the row plus 1 minus cup we want to sow. Then we deduct this from the number of seeds in the cup we want to sow. After that, we deduct 5 so many times, so that we get number between 1 and 5. After these operations we get number of the cup, where the sowing ends.
Practical example:
Again, we want to sow cup B4, which contains 12 seeds.
We count:
12 seeds - 2 (opposite to our cup 4 is opponent's cup number 2) or 12 seeds - ((5+1) - 4). (5 cups in the row, we sow cup 4, and 1 is always added)
12 - 2 = 10.
It is more than 5, so we deduct 5.
10 - 5 = 5.
Now we have number between 1 and 5, concretly 5. So the sowing will end up in cup 5.

This works for any size of the board. So let's try on standard board with nine cups.

T9T8T7T6T5T4T3T2T1
 
 
B1B2B3B4B5B6B7B8B9


We want to sow cup B4, which contains 32 seeds.
We count:
32 seeds + 4 (cup 4) - 1.
32 + 4 - 1 = 35.
It is more than 9 (we have 9 cups in the row), so we deduct 9.
35 - 9 = 26.
It is still more than 9, so we deduct 9 once more.
26 - 9 = 17.
It is still more than 9, so we deduct 9 once more.
17 - 9 = 8.
Now we have number between 1 and 9, concretly 8. So the sowing will end up in cup number 8.
Or the second method:
32 seed - 6 (opposite to cup 4 is cup 6).
32 - 6 = 26.
It is more than 9 (we have 9 cups in the row), so we deduct 9.
26 - 9 = 17.
It is still more than 9, so we deduct 9 once more.
17 - 9 = 8.
Now we have number between 1 and 9, concretly 8. So the sowing will end up in cup number 8.

This method does not say on which side the sowing ends, but it usually does not matter.

First block of

questions


1) On the board with five cups in the row we have cup number 1, which contains 20 seeds. Where do you put the last seed if you sow this cup?
Last seed will go to cup number
2) On the board with five cups in the row we have cup number 5, which contains 28 seeds. Where do you put the last seed if you sow this cup?
Last seed will go to cup number
3) On the board with five cups in the row we have cup number 2, which contains 15 seeds. Where do you put the last seed if you sow this cup?
Last seed will go to cup number
4) On the board with nine cups in the row we have cup number 2, which contains 35 seeds. Where do you put the last seed if you sow this cup?
Last seed will go to cup number
5) On the board with nine cups in the row we have cup number 7, which contains 7 seeds. Where do you put the last seed if you sow this cup?
Last seed will go to cup number
6) On the board with nine cups in the row we have cup number 5, which contains 28 seeds. Where do you put the last seed if you sow this cup?
Last seed will go to cup number
7) On the board with nine cups in the row we have cup number 8, which contains 14 seeds. Where do you put the last seed if you sow this cup?
Last seed will go to cup number
8) On the board with nine cups in the row we have cup number 3, which contains 15 seeds. Where do you put the last seed if you sow this cup?
Last seed will go to cup number

That is all.

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