# Encrypt the 2 digits multiplication table in my mind

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!

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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!

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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

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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.

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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
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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
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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.

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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!

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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).

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Wow

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Great

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Amazing

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Yes difficult

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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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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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PS.
Extension of criss cross.

Karatsuba algorithm