Mathemagics

Any two-digit by 11

32 x 11 >> 3+2=5 >> 352

note: add two digits, put result between digits

99 x 11 > 9+9=18 >> 1089 (carry the 1)

Square any two-digit that ends in 5

note: multiply 1st digit by next higher digit, then put 25 on the end

35 x 35 >> 3x4=12 >> 1225

note: multiple any two-digit numbers that have 1st digit and 2nd digits add to 10

83 x 87 >> 8x9=72 >> 3x7=21 >> 7221

Subtraction

1241-587 >> 1241-600 = 641 >> 641+13 = 654

Day of the week for any January 1st in 21st century

January 1st 2030 Last two digits of year 30 * 25% >> round down >> 7 >> Add to year >> 30+7=37 >> Subtract biggest multiple of 7 >> 37-35 = 2 >> Day of the week >> 2 = Tuesday (note: subtract 1 from adder if leap year)

Two-digit addition

67 + 28 >> 67 + 20 = 87 + 8 = 95

Three digit addition

538+327 >> 538+300 = 838 + 20 = 858 + 7 = 865

Two-digit subtraction

86-25 >> 86-20 = 66 - 5 = 61

Three-digit subtraction

958-417 >> 958-400 = 558 - 10 = 548 - 7 = 541

725 - 468 >> 725 - 500 = 225 + 32 = 257

2-by-1 multiplication

42 x 7 >> (40 x 7 = 280) + (2 x 7 = 14) = 294

69 x 6 >> (60 x 6 = 360) + (9 x 6 = 54) = 414 OR (70 x 6 = 420) - (1 x 6 = 6) = 414

3-by-1 multiplication

326 x 7 >> (300 x 7 = 2100) + (20 x 7 = 140) + (6 x 7 = 42) = 2282

Squaring two-digits

note: round up or down to nearest 10

13^2 >> (13+3=16) x (13-3=10) = 160+3^2 = 169

41^2 = (41+1=42) x (41-1=40) = 1680+1^2 = 1681

(b + c) x a = (b x a) + (c x a)

(b + c + d) x a = (b x a) + (c x a) + (d x a)

A^2 = (A + d) x (A - d) + d^2

(z + d)^2 = z x (z + 2 x d) + d^2

2-by-2 multiplication

46 x 42 >> (46 x 40 = 1840) + (46 x 2 = 92) = 1932

note: break up number with smaller last digit

59 x 17 >> (60 x 17 = 1020) - (1 x 17 = 17) = 1003

88 x 76 >> (90 x 76 = 6840) - (2 x 76 = 152) >> 6840 - 152 >> (long) 6840 - 200 + 48 = 6688 >> (short) 6600 + (complement(52 - 40 = 12) = 88) = 6688

2-by-2 multiplication where one number can be factored into one-digit numbers

Ex 42 = 7 x 6, 63 = 9 x 7, 84 = 7 x 6 x 2 or 7 x 4 x 3

46 x 42 >> (46 x 7 = 322) x 2 = 644 x 3 = 1932

Friendly products >> numbers w/ 0 in the center, easier to multiply

43 x 56 = 43 x 8 x 7 = 344 x 7 = 2408 (non-friendly)

43 x 56 = 43 x 7 x 8 = 301 x 8 = 2408 (friendly)

note: list of friendly product numbers

12x9=108 13x8=104 15x7=105 17x6=102 18x6=108 21x5=105 Etc

note: numbers that are factors of 11, use earlier trick

52 x 33 = 52 x 11 x 3 = 572 x 3 = 1716 83 x 66 = 83 x 11 x 6 = 913 x 6 = 5478

Three-digit squares

note: round up or down to nearest 100

193^2 >> (200 x 186) + 7^2 = 37200 + 49 = 37249

706^2 >> (700 x 712) + 6^2 = 498400+36 = 498436

note: farther away from 100, harder it gets

529^2 = (558 x 500) + 29^2 = 279000 + 29^2 >> 29^2 = (30 x 28) + 1^2 = 841 >> 279841

636^2 = (672 x 600) + 36^2 = 403200 + 36^2 >> 36^2 = (42 x 30) + 6^2 = 1260+36=1296 >> 404496

Cubing

A^3 = (A - d) x A x (A + d) + (d^2) x A

13^3 = (10 x 13 x 16) + (3^2 x 13)

13 x 16 = 13 x 4 x 4= 52 x 4 = 208

9 x 13 = 117

13^2 = 2080 + 117 = 2197

35^2 = (30 x 35 x 40) + (5^2 x 35) = (30 x 1400) + (35 x 5 x 5) = 42000 + (175 x 5) = 42000 + 875 = 42875