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Introduction

As we have seen in the previous chapter, the "123456789 exercise", consisting of adding that number nine times to a cleared abacus until reaching the number 1111111101 and then subtracting it nine times until the abacus is cleared again, has been used since ancient times to illustrate and practice addition and subtraction. It is a convenient exercise because:

• it is long enough to be a non-trivial exercise
• if we do not return to the initial value (zero) it is a sign that we have made a mistake
• we do not need a book or exercise sheet
• uses many of the elementary cases of addition or subtraction of a digit to/from another digit

but it also has a couple of drawbacks:

• it does not use all pairs of digits (ex. a 3 is never added to a 5)
• after repeating it several times, one begins to mechanically memorize the exercise, so that we are no longer practicing addition and subtraction

To avoid these two problems we can extend the exercise in several ways.

Using a background

Instead of using a cleared abacus, we can fill 9 columns with a digit (111111111, 222222222, etc.), this multiplies by 10 the number of exercises at our disposal. Now we are sure to use all possible cases of addition and subtraction digit by digit while mechanical memorization becomes harder.

The following table contains the intermediate values for reference. Such values are traversed from top to bottom in addition and from bottom to top in subtraction.

Table 1: The 123456789 exercise with background
+1..9 0 1 2 3 4 +1..9
0 000000000 111111111 222222222 333333333 444444444 0
1 123456789 234567900 345679011 456790122 567901233 1
2 246913578 358024689 469135800 580246911 691358022 2
3 370370367 481481478 592592589 703703700 814814811 3
4 493827156 604938267 716049378 827160489 938271600 4
5 617283945 728395056 839506167 950617278 1061728389 5
6 740740734 851851845 962962956 1074074067 1185185178 6
7 864197523 975308634 1086419745 1197530856 1308641967 7
8 987654312 1098765423 1209876534 1320987645 1432098756 8
9 1111111101 1222222212 1333333323 1444444434 1555555545 9
Table 1: The 123456789 exercise with background (continuation)
+1..9 5 6 7 8 9 +1..9
0 555555555 666666666 777777777 888888888 999999999 0
1 679012344 790123455 901234566 1012345677 1123456788 1
2 802469133 913580244 1024691355 1135802466 1246913577 2
3 925925922 1037037033 1148148144 1259259255 1370370366 3
4 1049382711 1160493822 1271604933 1382716044 1493827155 4
5 1172839500 1283950611 1395061722 1506172833 1617283944 5
6 1296296289 1407407400 1518518511 1629629622 1740740733 6
7 1419753078 1530864189 1641975300 1753086411 1864197522 7
8 1543209867 1654320978 1765432089 1876543200 1987654311 8
9 1666666656 1777777767 1888888878 1999999989 2111111100 9

The 987654321 exercise

Additionally, instead of using the number 123456789 we can think of using any permutations of these digits that we are able to comfortably remember; for example, 987654321, the only one we will consider here. This gives us 10 other independent exercises for addition and subtraction practice. The following table shows us the intermediate values of this new series of exercises using a background.

In total, we already have 20 different exercises.

Table 2: The 987654321 exercise with background starting with addition
+9..1 0 1 2 3 4 +9..1
0 000000000 111111111 222222222 333333333 444444444 0
1 987654321 1098765432 1209876543 1320987654 1432098765 1
2 1975308642 2086419753 2197530864 2308641975 2419753086 2
3 2962962963 3074074074 3185185185 3296296296 3407407407 3
4 3950617284 4061728395 4172839506 4283950617 4395061728 4
5 4938271605 5049382716 5160493827 5271604938 5382716049 5
6 5925925926 6037037037 6148148148 6259259259 6370370370 6
7 6913580247 7024691358 7135802469 7246913580 7358024691 7
8 7901234568 8012345679 8123456790 8234567901 8345679012 8
9 8888888889 9000000000 9111111111 9222222222 9333333333 9
Table 2: The 987654321 exercise with background starting with addition (continuation)
+9..1 5 6 7 8 9 +9..1
0 555555555 666666666 777777777 888888888 999999999 0
1 1543209876 1654320987 1765432098 1876543209 1987654320 1
2 2530864197 2641975308 2753086419 2864197530 2975308641 2
3 3518518518 3629629629 3740740740 3851851851 3962962962 3
4 4506172839 4617283950 4728395061 4839506172 4950617283 4
5 5493827160 5604938271 5716049382 5827160493 5938271604 5
6 6481481481 6592592592 6703703703 6814814814 6925925925 6
7 7469135802 7580246913 7691358024 7802469135 7913580246 7
8 8456790123 8567901234 8679012345 8790123456 8901234567 8
9 9444444444 9555555555 9666666666 9777777777 9888888888 9

Starting with subtraction

If you still do not have enough with the 20 previous exercises, you can count on another 20 independent exercises just start by subtracting 123456789 or 987654321 from the background nine times, after which we will return the abacus to its original state by adding the number nine times. In doing so, sooner or later we will find negative numbers that we can handle on "the other side" of the abacus; that is, by borrowing 1 from a real or imaginary column located further to the left. Before ending the exercise, that borrowed 1 will be returned with a carry to its real or imaginary column, and we will be able to finish the exercise with the abacus in its original state.

Table 3: The 123456789 exercise with background starting with subtraction
-1..9 0 1 2 3 4 -1..9
0 000000000 111111111 222222222 333333333 444444444 0
1 9876543211 9987654322 98765433 209876544 320987655 1
2 9753086422 9864197533 9975308644 86419755 197530866 2
3 9629629633 9740740744 9851851855 9962962966 74074077 3
4 9506172844 9617283955 9728395066 9839506177 9950617288 4
5 9382716055 9493827166 9604938277 9716049388 9827160499 5
6 9259259266 9370370377 9481481488 9592592599 9703703710 6
7 9135802477 9246913588 9358024699 9469135810 9580246921 7
8 9012345688 9123456799 9234567910 9345679021 9456790132 8
9 8888888899 9000000010 9111111121 9222222232 9333333343 9
Table 3: The 123456789 exercise with background starting with subtraction (continuation)
-1..9 5 6 7 8 9 -1..9
0 555555555 666666666 777777777 888888888 999999999 0
1 432098766 543209877 654320988 765432099 876543210 1
2 308641977 419753088 530864199 641975310 753086421 2
3 185185188 296296299 407407410 518518521 629629632 3
4 61728399 172839510 283950621 395061732 506172843 4
5 9938271610 49382721 160493832 271604943 382716054 5
6 9814814821 9925925932 37037043 148148154 259259265 6
7 9691358032 9802469143 9913580254 24691365 135802476 7
8 9567901243 9679012354 9790123465 9901234576 12345687 8
9 9444444454 9555555565 9666666676 9777777787 9888888898 9

Table 4: The 987654321 exercise with background starting with subtraction
-9..1 0 1 2 3 4 -9..1
0 000000000 111111111 222222222 333333333 444444444 0
1 9012345679 9123456790 9234567901 9345679012 9456790123 1
2 8024691358 8135802469 8246913580 8358024691 8469135802 2
3 7037037037 7148148148 7259259259 7370370370 7481481481 3
4 6049382716 6160493827 6271604938 6382716049 6493827160 4
5 5061728395 5172839506 5283950617 5395061728 5506172839 5
6 4074074074 4185185185 4296296296 4407407407 4518518518 6
7 3086419753 3197530864 3308641975 3419753086 3530864197 7
8 2098765432 2209876543 2320987654 2432098765 2543209876 8
9 1111111111 1222222222 1333333333 1444444444 1555555555 9

Table 4: The 987654321 exercise with background starting with subtraction (continuation)
-9..1 5 6 7 8 9 -9..1
0 555555555 666666666 777777777 888888888 999999999 0
1 9567901234 9679012345 9790123456 9901234567 12345678 1
2 8580246913 8691358024 8802469135 8913580246 9024691357 2
3 7592592592 7703703703 7814814814 7925925925 8037037036 3
4 6604938271 6716049382 6827160493 6938271604 7049382715 4
5 5617283950 5728395061 5839506172 5950617283 6061728394 5
6 4629629629 4740740740 4851851851 4962962962 5074074073 6
7 3641975308 3753086419 3864197530 3975308641 4086419752 7
8 2654320987 2765432098 2876543209 2987654320 3098765431 8
9 1666666666 1777777777 1888888888 1999999999 2111111110 9

This is the most interesting proposal in the context of traditional methods. The forty exercises above can be performed using the lower 5th bead, as explained in detail in the previous chapter: Use of the 5th lower bead, which will allow you to master this traditional technique.

With this, we have a total of 80 exercises!

Using alternate operation

And finally, why not? Even if only for the pleasure of overcoming a different difficulty, we can combine the previous exercises with an alternating direction of operation, from left to right and from right to left, as explained in the introductory chapter to addition and subtraction.

Example
Abacus Comment
ABCDEFGHIJ
Cleared abacus
+1
+2
+3
+4
+5
+6
+7
+8
+9
123456789 First step completed
+9
+8
+7
+6
+5
+4
+3
+2
+1
246913578 Second step completed
etc.

With this, you could go one step further in your understanding of bead mechanics.

Conclusion

With the 160 exercises presented here, you no longer have an excuse, you can practice addition and subtraction for hours, without exercise sheets, while comfortably seated on your sofa with only your abacus resting on your knees.

This is a door to mastery!

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