I.
THE SQUARE OF A NUMBER IS ONE GREATER THAN THE PRODUCT OF THE INTEGERS ADJACENT TO THAT NUMBER, THUS:

X² = 1 + (X+1) (X-1)

EX:
7²  = 1 + (8) (6)
9²  = 1 + (10) (8)

II.
THE PRODUCT OF TWO CONSECUTIVE NUMBERS IS TWO GREATER THAN THE PRODUCT OF THE INTEGERS ADJACENT TO THOSE NUMBERS, THUS:

(X) (X+1) = 2 + (X-1) (X+2)

EX:
(7)  (8)      = 2 + (6) (9)
(4)  (5)      = 2 + (3) (6)

III.
THE PRODUCT OF TWO CONSECUTIVE EVEN OR ODD NUMBERS IS THREE GREATER THAN THE PRODUCT OF THOSE INTEGERS ADJACENT TO THOSE NUMBERS, THUS:

(X) (X+2)  = 3 + (X-1) (X+3)

EX:
(6)  (8)       = 3 + (5) (9)
(3)  (5)       = 3 + (2) (6)

IV.
SIMILARLY, THE PRODUCT OF ANY TWO INTEGERS IS ALWAYS ONE GREATER THAN THE DIFFERENCE BETWEEN THOSE INTEGERS PLUS THE PRODUCT OF THE INTEGERS IMMEDIATELY LOWER THAN THE LOWER NUMBER AND HIGHER THAN THE HIGHER NUMBER, THUS:

(X) (Y)   = {1 + (Y-X)} +  {(X-1) (Y+1)}

EX:
(X) (X+9)  = 10 + (X-1) (X+10)
(6) (15)     = 10 + (5) (16)
(7) (16)     = 10 + (6) (17)
(8) (17)     = 10 + (7) (18)

EX:
(X) (X+22) = 23 + (X-1) (X+23)
(7) (29)      = 23 + (6) (30)
(8) (30)      = 23 + (7) (31)
(9) (31)      = 23 + (8) (32)