- WELCOME TO A LESSON ON INCA COUNTING BOARDS.
TO KEEP GOOD RECORDS,
THE INCAS USED QUIPUS TO RECORD QUANTITIES OF ITEMS.
A QUIPU IS A COLLECTION OF CORDS WITH KNOTS IN THEM,
WHICH WE WILL DISCUSS MORE IN THE NEXT LESSON.
BEFORE MAKING RECORDS, THE INCAS OFTEN NEEDED TO DO COMPUTATIONS.
TO DO THESE COMPUTATIONS,
THEY WOULD SOMETIMES USE A COUNTING BOARD
CONSTRUCTED WITH A SLAB OF STONE.
IN THE SLAB WERE CUT RECTANGULAR AND SQUARE COMPARTMENTS
SO THAT AN OCTAGONAL, OR EIGHT-SIDED REGION,
WAS LEFT IN THE MIDDLE, WHICH WOULD BE THIS REGION HERE.
TWO OPPOSITE CORNER RECTANGLES WERE RAISED,
WHICH WOULD BE THIS RECTANGLE AND THIS RECTANGLE.
ANOTHER TWO SECTIONS WERE MOUNTED
ON THE ORIGINAL SURFACE OF THE SLAB
SO THAT THERE WERE ACTUALLY THREE LEVELS AVAILABLE.
IN THE FIGURE SHOWN BELOW, THE DARK[ISH] SHADED CORNER REGIONS
REPRESENTED THE HIGHEST THIRD LEVEL,
WHICH WOULD BE THIS SQUARE AND THIS SQUARE.
THE LIGHTER SHADED REGIONS [STARTING IN] THE CORNERS
ARE THE SECOND HIGHEST LEVELS,
WHILE THE CLEAR WHITE RECTANGLES AND SQUARES ARE THE COMPARTMENTS
CUT INTO THE STONE SLAB.
PEBBLES WERE USED TO KEEP ACCOUNTS AND THEIR POSITIONS
WITHIN THE VARIOUS LEVELS AND COMPARTMENTS [GAVE] TOTALS.
FOR EXAMPLE, A PEBBLE IN A SMALL WHITE COMPARTMENT
REPRESENTED ONE UNIT.
SO LOOKING AT OUR DIAGRAM HERE,
ALL THESE COMPARTMENTS WOULD BE WORTH ONE.
IF A PEBBLE WAS PUT INTO ONE OF THE TWO
WHITE LARGER RECTANGULAR COMPARTMENTS,
ITS VALUE WAS DOUBLED.
SO PEBBLES IN THIS COMPARTMENT AND THIS COMPARTMENT
WERE DOUBLED, OR WORTH TWO EACH.
WHEN A PEBBLE WAS PUT INTO THE OCTAGONAL REGION
IN THE MIDDLE OF THE SLAB,
ITS VALUE WAS TRIPLED, WHICH WOULD BE THIS REGION HERE.
IF THE PEBBLE WAS PLACED ON THE SECOND SHADED LEVEL,
ITS VALUE WAS MULTIPLIED BY SIX--
THAT WOULD BE HERE AND HERE.
AND FINALLY, IF A PEBBLE WAS FOUND
IN ONE OF THE TWO HIGHEST CORNER LEVELS,
ITS VALUE WAS MULTIPLIED BY 12-- THAT WOULD BE HERE AND HERE.
DIFFERENT OBJECTS COULD BE COUNTED AT THE SAME TIME
BY REPRESENTING DIFFERENT OBJECTS
BY DIFFERENT COLORED PEBBLES.
LET'S LOOK AT AN EXAMPLE.
SUPPOSE A CIRCLE REPRESENTS ONE DOG
AND A SQUARE REPRESENTS ONE GOAT--
DETERMINE THE TOTAL SHOWN ON THE COUNTING BOARD.
LET'S BEGIN BY LABELING THE VALUE OF EACH COMPARTMENT.
REMEMBER, THE SMALLEST COMPARTMENTS IN THE WHITE REGION
ARE WORTH ONE EACH,
AND THERE ARE A TOTAL OF 12 OF THESE COMPARTMENTS.
NEXT, THESE TWO LARGER WHITE RECTANGLES ARE WORTH TWO EACH--
HERE AND HERE.
THE EIGHT-SIDED COMPARTMENT IN THE MIDDLE IS WORTH THREE EACH.
AND IN THE CORNERS, THE SECOND HIGHEST LEVELS
ARE WORTH SIX EACH-- HERE AND HERE.
AND THEN THE HIGHEST LEVEL IS WORTH 12 EACH--HERE AND HERE.
LET'S BEGIN BY DETERMINING THE VALUE OF THE RED CIRCLES,
WHICH REPRESENT THE NUMBER OF DOGS.
NOTICE HOW THERE ARE TWO PEBBLES IN THIS COMPARTMENT HERE
WITH 12 EACH.
SO WE'D HAVE 2 x 12 + WE HAVE TWO PEBBLES HERE WORTH SIX EACH,
SO 2 x 6 + LOOKING AT THE COMPARTMENT IN THE CENTER
THESE TWO PEBBLES ARE WORTH THREE EACH,
SO WE HAVE 2 x 3 + WE HAVE THREE PEBBLES HERE WORTH TWO EACH,
SO 3 x 2.
NOW WE'RE DOWN TO THE COMPARTMENTS
THAT ARE WORTH ONE EACH.
THERE ARE THREE RED PEBBLES HERE, SO WE HAVE + 3 x 1.
SO WE HAVE A TOTAL OF 24 + 12 + 6 + 6 + 3.
SO WE HAVE 24 + 12, THAT'S 36, + 6, THAT'S 42,
+ 6 IS 48 + 3 IS 51.
AND EACH RED CIRCLE REPRESENTS ONE DOG,
SO WE HAVE 51 TOTAL DOGS.
NOW LET'S DETERMINE THE VALUE OF THE BLUE SQUARES,
WHICH REPRESENTS THE NUMBER OF GOATS.
AGAIN, LET'S START AT THE HIGHEST COMPARTMENTS,
OR THE COMPARTMENTS WORTH THE MOST.
HERE WE HAVE ONE PEBBLE WORTH 12, SO 1 x 12,
+ THESE THREE PEBBLES ARE WORTH SIX EACH, SO WE HAVE 3 x 6,
+ THERE AREN'T ANY BLUE SQUARES IN THIS COMPARTMENT HERE,
WHERE EACH PEBBLE IS WORTH THREE,
SO WE'LL GO TO THE DEPARTMENTS WORTH TWO--
THIS COMPARTMENT AND THIS COMPARTMENT.
NOTICE HOW THERE'S ONE BLUE PEBBLE HERE AS WELL AS ONE HERE.
SO WE HAVE TWO PEBBLES WORTH TWO EACH.
SO WE'LL FINALLY LOOK AT THE COMPARTMENTS THAT ARE WORTH ONE.
WE HAVE A TOTAL OF THREE BLUE SQUARES HERE WORTH ONE EACH,
SO +3 x 1.
SO WE HAVE A TOTAL OF 12 + 18 + 4 + 3, SO -- 37.
SO THERE'S 37 IN TOTAL GOATS.
OKAY, I HOPE YOU FOUND THIS HELPFUL.