# Traditional Abacus and Bead Arithmetic/Addition and subtraction/Extending the 123456789 exercise

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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 |

## Using the 5th lower bead

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!