Encrypt the 2 digits multiplication table in my mind

[Antelex]

In the recent month, I was trying to use mnemonics and math equations to boost my 2 digits multiplication skills. I am glad that I did a decent improvement (a month ago ~8sec per question to now ~4s/q) though it’s still far from satisfactory (~2s/q). Anyway。。。

What I really want to tell you is the structuralised multiplication table, I had recited. At first, I will clarify what combinations I considered worthless to recite. Then, I will talk about the math tricks I used such that these combinations are also redundant to be memorised. Finally, it will be my structuralised multiplication table to archive PB 4s/q under multiple times of 2 minutes unlimited questions tested. Enjoy reading!

2DMT.csv (103.0 KB) Free, not for sale!

[Antelex]

You have roted
Given that you are super familiarized with 9x9 multiplication table.
Then, any digits ends up with zero are under 9x9T, eg 90x80=9x8x100=7200 or 70x7=7x7x10=490。。。

Next step train with 2 digits times one digits, it means 48x8=40x8+8x8=320+64=384 or 62x7=60x7+2x7=434。。。
Unless, you can say out the answer within a second, keep training!

[Antelex]

Math tricks used
In this round, we will focus on the combinations that can be simplified by tricks.

1. x11
   34x11 → 3_4 → 3+4=7 → 374
   67x11 → 6_7 → 6+7=13 → 737.

2. x37
   Given that 37x3=111, 37xmultipliers_of_3。。。
   eg. 37x6 → 6/3=2 → 222
   37x27 → 27/3=9 → 999
   37x66 → 66/3=22 → 2__2 → 2+2=4 → 2442.
   48x37 → 48/3=16 → 1__6 → 1+6=7 → 1776

3. x15 means one and a half
   38x15 → 38/2=19 → 380+190=570.
   79x15 → 79/2=39.5 → 790+395=1185.

4. x25 means divided by 4
   52x25=a deck of card 52 cards, one suit is 13, which is also a quarter of the deck, then 1300.
   48/25 → 48/4x100 → 1200
   Beware of the following patterns,
   remainder = 0 → 00, 1 → 25, 2 → 50, 3 → 75
   65x25 → 65/4=1x…25=16…1 → 1625
   66x25 → 66/4=1x…26=16…2 → 1650
   67x25 → 67/4=1x…27=16…3 → 1675# [long divisions]

5. x55 means half and half
   38x55 → [38/2=19] → 1900+190 → 2090
   87x55 → [87/2=43.5] → 4350+435 → 4785

6. units digit end with 5
   35x45 → floor(3x4 + [3+4=7]/2) = 15 → abs(3-4) → odd → 1575
   65x85 → floor(6x8 + [6+8=14]/2) = 55 → abs(6-8) → even → 5525

7. Square numbers
   Most of the patterns are stated in
   Square a 2 digits number within a second in mnemonic sense.

8. Double digits type a:
   33x26 → 3x2|[2+6=8]x3|3x6 → 6_24_18 → 858;
   42x88 → 4x8|[4+2=6]x8|2x8 → 32_48_16 → 3696;

9. Double digits type b:
   56x86 → 5x8|[5+8=13]x6|6x6 → 40_78_36 → 4816
   29x79 → 2x7|[2+7=9]x9|9x9 → 14_81_81 → 2291

10. Double digits type c:
    The tens digit is equal and the units digit add up to 10.
    66x64 → 6x[6+1=7]|6x4 → 4224
    23x27 → 2x[2+1=3]|3x7 → 621

11. units digit end in 9
    39x48 → 40x48 → 1920-48 → 1872
    69x24 → 70x24 → 1480-24 → 1456

[Antelex]

Structured Multiplication Table
Now, we have identified those combinations that we need not to memorized.
My following table tells the numbers are hard to be mental calculated.
criss cross is
_34
x56
3x5|3x6+4x5|4x6
15_38_24
1904

In criss cross, we have to identify the difficult combinations such we can increase our effectiveness of remembering.
34x56 is hard since we have to do addition with 1 carry.
15
_20
_18
__24
1904 carried 1 times
So if a multiplication needs addition with carries, it is told to be troublesome.
Best case zero carries, worst case 3 carries.

[Antelex]

Loci 1: Square

P_	A_	O1	O2	Remarks
33	10	89	68	33^2=1089 or 83^2=6889
34	11	56	70	34^2=1156 or 84^2=7056
36	12	96	73	…
37	13	69	75
38	14	44	77
39	15	21	79
71	50	41	62
72	51	84	60
73	53	29	59
74	54	76	57
62	38	44		62^2=3844
63	39	69		63^2=3969
66	43	56		…
67	44	89
68	46	24

[Antelex]

Loci 2: Consecutive

P1	P2	A	O1	O2	Remarks
22	23	05	06	52	22x23=506 or 23x24=552
26	27	07	02	56	26x27=702 or 27x28=756
33	34	11	22	90	33x34=1122 or 34x35=1190
37	38	14	06	82	37x38=1406 or 38x39=1482
42	43	18	06	92	42x43=1806 or 43x44=1892

P	A	O	Remarks
21	04	62	21x22=462
28	08	12	28x29=812
31	09	92	…
32	10	56
35	12	60	36x37 → 36/3=12 → 1332 (math tricks mentioned)
41	17	22
44	19	80
45	20	70
46	21	62
47	22	56
48	23	52
52	27	56
53	28	62	54x55,55x56 (trick)
56	31	92
57	33	06
58	34	22
61	37	82
62	39	06
63	40	32
64	41	60
65	42	90
66	44	22
67	45	56
68	46	92
71	51	12
72	52	56
73	54	02
74	55	50
75	57	00
76	58	52
77	60	06
78	61	62
81	66	42
82	68	06
83	69	72
84	71	40
85	73	10
86	74	82
87	76	56
88	78	32
91	83	72
92	85	56
93	87	42
94	89	30
95	91	20
96	93	12
97	95	06
98	97	02

[Antelex]

Why reciting square numbers?
By (A-B)(A+B) = A^2 - B^2,
we have learned a new math tricks!
E.g. When A=37 and B=1, (difference=2B)
we have (37-1)(37+1) = 36x38

1. Difference = 2
   36x38 → 37^2 - 1 → 1369 - 1 → 1368
   44x46 → 45^2 - 1 → 2024
   81x83 → 82^2 - 1 → 6723

2. Difference = 20
   36x56 → 46^2 - 100 → 2016
   59x79 → 69^2 - 100 → 3381

3. Difference = 4
   22x26 → 24^2 - 4 → 572
   87x91 → 89^2 - 4 → 7917

4. Difference = 40
   43x83 → 63^2 - 400 → 3569
   27x67 → 47^2 - 400 → 1809

It’s difficult to apply the trick when the Difference >= 6,
e.g. 69x75 → 75-69=6 (challenge 1 identification) → 78^2 - 9 (challenge 2 borrow is likely to happen when subtraction) → 6084-9.

[Antelex]

Why reciting consecutive numbers?
The answer is obvious, right?
By the same trick in the previous post, we can have Difference = 3, or 5.
But how to apply it, take 10 sec to think of it?

Alright, the answer is

1. Difference = 3
   Say 51x54
   (A-1.5)(A+1.5) → ??? → Wait we have already recited consecutive numbers!
   So it should be
   51 x 54 → (51+1)(54-1) - 2 → 52x53 - 2 → 2754
   Another example,
   68x71 → 69x70 - 2 → 4828

2. Difference = 30
   16x46 → 26x36 - 200 → 736

3. Difference = 5
   84x89 → (84+2)(89-2) - 6 → 86x87 - 6 → 8476

4. Difference = 50
   34x84 → 54x64 - 600 → 2856

Now, you might notice for Difference = 30 and 50, you don’t have the table.
That is TBC. ʕ•́ᴥ•̀ʔっThanks!

[Antelex]

Loci 3: Consecutive 2
|P|A|O1|O2|Remarks|
| — | — | — | — | — | — |
|12|02|64|44|12x22=264,62x72=4464
|22|07|04|59|22x32=704,72x82=5904
|32|13|44|75|32x32=1344,82x92=7544
|42|21|84||42x52=2184
|52|32|24||52x62=3224
|13|02|99|45|…
|23|07|59|60|
|33|14|19|77|
|43|22|79||
|53|33|39||
|14|03|36|47|
|24|08|16|62|
|34|14|96|78|
|44|23|76||
|54|34|56|?5x?5 → math tricks|
|16|04|16|50|
|26|09|36|65|
|36|16|56|82|
|46|25|76||
|56|36|96||
|17|04|59|51|
|77|66|99|27x37 → math tricks|
|37|17|39|84|
|47|26|79||
|57|38|19||
|18|05|04|53|
|28|10|64|68|
|38|18|24|86|
|48|27|84||
|58|39|44||
|19|05|51|54|
|29|11|31|70|
|39|19|11|88|
|49|28|91||
|59|40|71||
/] ( ≖.≖) /} Next time is about remembering some (2 digits x 1 digits).

[Ranipdo (Ranipari)]

Wow

[Ranipdo (Ranipari)]

Great

[Ranipdo (Ranipari)]

Amazing

[Ranipdo (Ranipari)]

Yes difficult

[Antelex]

Loci 4: 2 digits x 1 digit (Factor clustering)
Say 32x32=64x16=1024.
Similarly in 2 digits x 1 digits there are also many such cases, if we can react faster to different factors I can achieve better speed ~0.5sec.
Indeed, doing something like 39x46=40x46-46=1840-46, I can also instant recall 1840.
In short, by remembering a result and instant recall multiple combinations, it’s worthy not?

P x A = O
3 36 | 4 27 | 108
4 33 | 6 22 | 132
3 46 | 6 23 | 138
3 48 | 4 36 | 144
4 39 | 6 26 | 156
4 42 | 6 28 |168
4 46 | 8 23 | 184
3 63 | 7 27 | 189
4 48 | 6 32 | 192
4 49 | 7 28 | 196
3 66 | 6 33 | 9 22 | 198
3 69 | 9 23 | 207
3 72 | 6 36 | 8 27 | 216
7 32 | 8 28 | 224
3 76 | 6 38 | 228
3 77 | 7 33 | 231
3 78 | 6 39 | 9 26 | 234
4 63 | 6 42 | 7 36 | 9 28 | 252
3 86 | 6 43 | 258
3 87 | 9 29 | 261
3 88 | 4 66 | 8 33 | 264
3 92 | 4 69 | 6 46 | 276
3 96 | 4 72 | 6 48 | 8 36 | 9 32 | 288
3 98 | 6 49 | 7 42 | 294
3 99 | 9 33 | 297
4 76 | 8 38 | 304
4 78 | 8 39 | 312
7 48 | 8 42 | 336
4 86 | 8 43 | 344
4 92 | 8 46 | 368
4 93 | 6 62 | 372
6 63 | 9 42 | 378
4 96 | 8 48 | 384
4 98 | 8 49 | 392
4 99 | 6 66 | 396
6 69 | 9 46 | 414
6 72 | 9 48 | 432
7 63 | 9 49 | 441
6 77 | 7 66 | 462
7 72 | 8 63 | 504
6 88 | 8 66 | 528
6 92 | 8 69 | 552
6 93 | 9 62 | 558
6 96 | 8 72 | 576
6 99 | 9 66 | 594
7 88 | 8 77 | 616
7 99 | 9 77 | 693
8 99 | 9 88 | 792

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[Antelex]

Multiplication that involves 2 carries, also No Math tricks! So these combinations are said to be the most difficult problem set.
These loci are where I am working hard at.

Loci 5: 2 carries? Never see you again!
164 locations needed or can be group by person such that only ~40 locations need.
[P, A, OO]
[13, 85, 1105],
[13, 86, 1118],
[13, 87, 1131],
[14, 86, 1204],
[14, 87, 1218],
[16, 75, 1200],
[16, 78, 1248],
[16, 82, 1312],
[16, 83, 1328],
[16, 84, 1344],
[16, 85, 1360],
[16, 86, 1376],
[16, 87, 1392],
[17, 65, 1105],
[17, 68, 1156],
[17, 72, 1224],
[17, 73, 1241],
[17, 74, 1258],
[17, 75, 1275],
[17, 76, 1292],
[17, 78, 1326],
[17, 83, 1411],
[17, 84, 1428],
[17, 85, 1445],
[17, 86, 1462],
[17, 87, 1479],
[18, 62, 1116],
[18, 63, 1134],
[18, 64, 1152],
[18, 65, 1170],
[18, 67, 1206],
[18, 73, 1314],
[18, 74, 1332],
[18, 75, 1350],
[18, 76, 1368],
[18, 84, 1512],
[18, 85, 1530],
[18, 86, 1548],
[18, 87, 1566],
[23, 48, 1104],
[24, 46, 1104],
[24, 47, 1128],
[24, 48, 1152],
[26, 43, 1118],
[26, 47, 1222],
[26, 48, 1248],
[26, 82, 2132],
[26, 83, 2158],
[26, 84, 2184],
[26, 85, 2210],
[26, 87, 2262],
[27, 42, 1134],
[27, 43, 1161],
[27, 45, 1215],
[27, 46, 1242],
[27, 48, 1296],
[27, 82, 2214],
[27, 83, 2241],
[27, 84, 2268],
[27, 86, 2322],
[28, 43, 1204],
[28, 47, 1316],
[28, 75, 2100],
[28, 76, 2128],
[28, 83, 2324],
[28, 84, 2352],
[28, 86, 2408],
[28, 87, 2436],
[34, 62, 2108],
[34, 63, 2142],
[34, 65, 2210],
[34, 68, 2312],
[35, 63, 2205],
[36, 62, 2232],
[36, 63, 2268],
[36, 64, 2304],
[36, 65, 2340],
[36, 67, 2412],
[36, 68, 2448],
[36, 87, 3132],
[37, 58, 2146],
[37, 63, 2331],
[37, 65, 2405],
[37, 68, 2516],
[37, 84, 3108],
[37, 85, 3145],
[37, 86, 3182],
[38, 56, 2128],
[38, 62, 2356],
[38, 63, 2394],
[38, 64, 2432],
[38, 65, 2470],
[38, 67, 2546],
[38, 82, 3116],
[38, 83, 3154],
[38, 85, 3230],
[38, 86, 3268],
[38, 87, 3306],
[42, 74, 3108],
[45, 72, 3240],
[45, 74, 3330],
[45, 76, 3420],
[45, 78, 3510],
[46, 68, 3128],
[46, 72, 3312],
[46, 74, 3404],
[47, 72, 3384],
[47, 73, 3431],
[47, 74, 3478],
[47, 75, 3525],
[47, 76, 3572],
[47, 78, 3666],
[48, 65, 3120],
[48, 67, 3216],
[48, 72, 3456],
[48, 73, 3504],
[48, 74, 3552],
[48, 75, 3600],
[48, 76, 3648],
[48, 86, 4128],
[48, 87, 4176],
[54, 76, 4104],
[54, 78, 4212],
[56, 75, 4200],
[57, 72, 4104],
[57, 73, 4161],
[57, 74, 4218],
[57, 75, 4275],
[57, 76, 4332],
[57, 78, 4446],
[58, 73, 4234],
[58, 75, 4350],
[58, 76, 4408],
[62, 83, 5146],
[62, 84, 5208],
[62, 86, 5332],
[63, 86, 5418],
[64, 83, 5312],
[64, 86, 5504],
[67, 78, 5226],
[67, 83, 5561],
[67, 84, 5628],
[67, 86, 5762],
[68, 75, 5100],
[68, 76, 5168],
[68, 83, 5644],
[68, 84, 5712],
[68, 86, 5848],
[68, 87, 5916],
[72, 85, 6120],
[72, 87, 6264],
[73, 84, 6132],
[73, 85, 6205],
[73, 87, 6351],
[74, 87, 6438],
[75, 84, 6300],
[75, 87, 6525],
[76, 83, 6308],
[76, 84, 6384],
[76, 85, 6460],
[76, 87, 6612],
[78, 85, 6630],
[78, 86, 6708],
[78, 87, 6786].
Hopefully, once mastered such a question, can speed up to around 3 sec per 2 digits multiplication. And my goal to 2 sec is another story.
It is the end of all my research and analysis, thanks for your participation and patient.
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───▄▄██▌█░░Full Of Mem░░▐<<—==
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▀❍▀▀▀▀▀▀▀❍❍▀▀▀▀▀▀❍❍▀=======

[Antelex]

PS.
Extension of criss cross.

Karatsuba algorithm

Also apply for higher degree of multiplicaion:
eg. 1234x5678
=12x56|12x78+34x56|34x78
=672|2840|2652
=672
+_2840
+____2652
=7006652

[Rishav123q (RISHAV PRATAP SINGH)]

dear brother,
i am kinda new to this can you please explain a bit what is this .cause i am very keen to learn tables upto 50

[Antelex]

welcome rishav
I thought this thread is long cold and closed.
what makes this technique special was that I didn’t mean to rote the whole table. The main aim of this topic is that
we use mnemonics to learn the 2 digits multiplication table by patterns. Then converted it into muscle memory.

[Rishav123q (RISHAV PRATAP SINGH)]

Brother you can contact me. As I need your help bro

[Antelex]

sorry rishav, I don’t do private coaching, also I am not a memory athlete and I don’t have any memory record.

To remind you that if you post your email and phone call here, it will become public. please consider if it is what you want.