Many Egyptian problems have more than one solution. You should not automatically assume that if your answers don’t look like the following solutions that they are wrong. If you have a calculator, find a decimal approximation for your and my solutions and see if they are the same.
PRACTICE: Add GGG DDDDDDA + GGGGGGGFFDDDDDDDAA.
ANSWER: HFFFDDDAAA
PRACTICE: Multiply 13 by 12 as an Egyptian would.
PRACTICE: Divide 187 by 17 and 100 by 21.
PRACTICE: Divide 2 by 5 using “sliced bread.” Make your first cut in thirds.
ANSWER: d ag
PRACTICE: Divide 4 by 18 using “sliced bread.” Make your first cut in fifths.
ANSWER: g fg
PRACTICE: Divide 5 by 7 in two ways using “sliced bread.”
or
ANSWER: s h sf ahk or s j af
PRACTICE: Find half of 58.
ANSWER: 29
PRACTICE: Find half of 258 by breaking up the number.
ANSWER: 129
PRACTICE: Find half of 8743 by breaking up the number.
ANSWER: 4371 s
PRACTICE: Repeat for 951.
ANSWER: 475 s
PRACTICE: Find the area of a triangle that has a base of 9 and a height of 10.
ANSWER: 45
PRACTICE: Repeat for a base of 23 and a height of 11.
ANSWER: 126 s
PRACTICE: Find the circumference of a circle of diameter 14.
ANSWER: 43 s f
PRACTICE: Repeat for diameter 31.
ANSWER: 96 s f k
PRACTICE: Multiply 48 by 1 k ds.
ANSWER: 55 s
PRACTICE: Multiply 3 s; by 6.
ANSWER: 18 g a;
PRACTICE: Multiply 6 af by 3 f.
ANSWER: 19 s j af gh
PRACTICE: Multiply 3 h; by 16.
ANSWER: 48 g ag
PRACTICE: How many palms would you need to push a 3 f-cubit block back if the seked were 7?
ANSWER: 22 s f palms
PRACTICE: Divide 23 by 10
ANSWER: 2 g a;
PRACTICE: Divide 68 by 12.
ANSWER: 5 ' or 5 s h
PRACTICE: Is a pyramid with height of 24 and a base of 52 safe?
Run = half 52 = 26, Rise = 24
ANSWER: Yes, since its seked, 7 d h as, is greater than 5 s.
PRACTICE: What is ' of 15?
SOLUTION: Since 10 + 5 = 15, ' of 15 is 10.
ANSWER: 10
PRACTICE: Find ' of 24 and 39,156.
SOLUTION: Since 16 + 8 = 24, ' of 24 is 16.
'×(30,000 + 9,000 + 150 + 6)
= 20,000 + 6,000 + 100 + 4 = 26,104
ANSWERS: 16 and 26,104
PRACTICE: Find ' of 192 and 294.
SOLUTION: ' × 192 = '(180 + 12) = 120 + 8 = 128
' × 294 = '(270 + 24) = 180 + 16 = 196.
ANSWERS: 128 and 196
PRACTICE: Find ' of 36,168 and 574,329.
SOLUTION: ' × 36,168 = '(30,000 + 6,000 + 150 + 18)
= 20,000 + 4,000 + 100 + 12 = 24,112 ' × 574,329
= '(300,000 + 270,000 + 3,000 + 1,200 + 120 + 9)
= 200,000 + 180,000 + 2,000 + 800 + 80 + 6
= 382,886
PRACTICE: Find ' of 9,454 and 41,627.
SOLUTION: ' × 9,454 = '(9,000 + 300 + 150 + 3 + 1) = 6,000 + 200 + 100 + 2 + ' = 6,302 ' ' × 41,627 = '(30,000 + 9,000 + 2,400 + 210 + 15 + 2) = 20,000 + 6,000 + 1,600 + 140 + 10 + 1 d = 27,751 d
ANSWERS: 6302 ' and 27,751 d
PRACTICE: Find ' of d;.
SOLUTION: ' × d; = 
= fg
ANSWER: fg
PRACTICE: Take ' of aa.
SOLUTION: ' × aa = 
= ss hh
ANSWER: ss hh
PRACTICE: If a 1 share is g s;, find the night watchman’s share.
SOLUTION: ' × g s; = 
= a; d; d; = a; ag
ANSWER: a; ag
PRACTICE: If a 1 share is ' h, find the night watchman’s share.
SOLUTION: ' × ' h = d l l = d h ak
ANSWER: d h ak, or equivalently, s ak
PRACTICE: Divide the following by 3 using the Egyptian trick.
• 24
• 52
• 35 l
ANSWERS: 8, 17 d, and 11 ' dh a;k
PRACTICE: Divide 9 and 27 s by 18.
PRACTICE: Find the volume of a 3-by-4-by-5 block and a 7-by-3-by-1 s block.
ANSWERS: 60 and 31 s
PRACTICE: What is 53 s da ÷ 10?
SOLUTION: 53 s da ÷ 10 = (50 + 3 + s + da) ÷ 10
= 5 + g a; + s; + da;
= 5 g a; s; da;
ANSWER: 5 g a; s; da;
PRACTICE: Bricks are piled in a 3-by-2 s-by-7 cubic cubit block. How many enlistees are needed to load the bricks?
SOLUTION: Volume = 3 × 2 s × 7.
Number of enlistees = volume ÷ 10
52 s ÷ 10 = (50 + 2 + s) ÷ 10 = 5 g s;
ANSWER: 5 g s; or 5 f
PRACTICE: Use inches and feet to simplify d f as as feet.
d f as feet = 4 + 3 + 1 inches
= 8 inches
= ' feet
ANSWER: ' feet
PRACTICE: Use dollars and pennies to simplify s; sg g; dollars.
s; sg g; dollars = 5 + 4 + 2 pennies
= 11 pennies
= 10 + 1 pennies
= a; a;; dollars
ANSWER: a; a;;
PRACTICE: Add ', g, h, and d; as parts of 30.
ANSWER: 1 ag
PRACTICE: Add f, g, a;, and s; as parts of 20.
PRACTICE: Simplify the following:
as dh
s; ak;
SOLUTION:
ANSWER: l, ak
PRACTICE: Multiply and use the G rule to simplify your answer.
5 × s;
7 × sf
ANSWER:
f
f sf or h k
or
PRACTICE: Multiply then simplify using j af sk = f
5 × j sk
5 f × j
ANSWER:
s f j
s f
PRACTICE: Multiply then simplify using ak sj gf = l
dh gf × 6
PRACTICE: Simplify as ak dh.
SOLUTION: Since 12 × 1 s = 18 and 18 × 2 is 36, as ak dh = 12 ÷ 2 = h.
ANSWER: h
PRACTICE: Make an identity for the simplification of g a; d;.
ANSWER: d
PRACTICE: Using the whole parts of 30, find all identities that sum to d.
d is 10 parts of 30. Since 6 + 3 + 1 and 5 + 3 + 2 both equal 10, we get g a; d; and h a; ag.
ANSWER: g a; d; and h a; ag
PRACTICE: Using the whole parts of 30, find expressions for 3 ÷ 10 and 8 ÷ 10.
ANSWER: g a; and ' a; d;
PRACTICE: Find an expression for 2 × ag using the factors 3 and 1.
SOLUTION:
ANSWER: a; d;
PRACTICE: Find an expression for 2 × ag using the factors 15 and 5.
ANSWER: a; d;
PRACTICE: Using algebra, prove that 3 × as = f.
PROOF: 3 × as = 3 × 1/12 = 3/1 × 1/12 = 3×1/1×12 = 3/12 = 1/4 = f
PRACTICE: Prove 
PROOF: 
PRACTICE: Add
8 ' ag as; + 2 a; as; + 9 d ag.
(8 + 2 + 9) + (' d) a; (ag ag) (as; as;)
= 19 + 1 (a; a;) (d; h;)
= 20 (g s;)
= 20 f
ANSWER: 20 f
PRACTICE: Build a gh in the first column when the first row is j, 12 g.
ANSWER: The fourth row should be gh, 1 s f;.
PRACTICE: Build a ag in the first column when the first row is g, 11 s.
ANSWER: The third row should be ag, 3 ' h.
PRACTICE: Build a sg in the first column when the first row is g, 7 f.
ANSWER: The third row should be sg, 1 d ag s;.
PRACTICE: Build a d;; in the first column given a first row of s;, 15 d.
ANSWER: The third row should be d;;, 1 fg.
PRACTICE: Build a dd in the first column given a first row of k, 11.
ANSWER: The fourth row should be dd, 2 '.
PRACTICE: Prove 2 ÷ 27 is ak gf.
PRACTICE: Prove 2 ÷ 17 is as ga hk.
PRACTICE: Simplify the following or explain why there is no need.
• g sa f;
21 is more than three times 5 so the first pair are fine. While 40 is roughly twice 21, it’s the second pair, so it is fine.
• d f dh
4 is much less than three times 3 so we will use parts of 36, the LCM.
So d f dh simplifies to s l.
• h s; sf
While 20 is more than three times 6, the 20 and 24 are too close. Simplify with parts of 120.
So h s; sf simplifies to f as;.
PRACTICE: Simplify d g so the solution starts with s.
ANSWER: s d;
PRACTICE: Try to simplify k ds starting with g. What goes wrong and why?
There’s nothing you can add to 32 to get 25.
PRACTICE: Find a good approximation for dd dj.
SOLUTION: The even numbers between the two are 34 and 36. 36 is as close to 37 as 33, so half, ak, is a decent approximation.
ANSWER: ak
PRACTICE: Consider the simplification of ak sa. Which is the best fraction to start with, aa, as, or ad? Explain and simplify with this fraction.
SOLUTION: as is the best since it has many small factors and shares a 3 with both ak and sa and a 2 with ak.
ANSWER: as gf jgh, as hd sgs, or as kf ash
PRACTICE: Add a; af as parts of 10 × 14. Use Egyptian multiplication tricks to figure out their parts and the parts of the final solution. Include a multiplication-like document to show their derivation.
SOLUTION:
ANSWER: j dg
PRACTICE: Construct Ahmose’s doubling of aa starting with the fraction h. Do all parts as a table.
SOLUTION:
ANSWER: h hh
PRACTICE: Repeat for ad starting with an k.
SOLUTION:
ANSWER: k gs a;f
PRACTICE: Find the area of a trapezoid with a height of 9 and bases of 10 and 13. Give the answer in cubits and cubit strips.
SOLUTION: Add the bases to get 23. Take half to get 11 s. Multiply 11 s by the height of 9 to get 100.
This is the answer in square cubits. Multiply by a;;, the number of cubit strips in a square cubit, to get the answer, f; a;;, in cubit strips.
1 g; a;; s;; = 1 (g; s;;) a;; = 1 f; a;;
ANSWER: 103 s cubits and 1 f; a;; cubit strips
PRACTICE: Starting with the given rows, create the last number on the right in four lines or less.
• 11, 90 to 3
• 1, 45 to 31
• 11, 18 to 45
• 12, 9 to dh
• f, 50 to 58
• 4, 150 to 5
• h, 80 to 89
• 10, 36 to 48
• 20, 12 to fg
PRACTICE: Complete f g to s.
ANSWER: s;
PRACTICE: Complete aa hh to h.
ANSWER: ss hh
PRACTICE: Complete d g to 1.
ANSWER: d a; d;
PRACTICE: Complete aa to 1.
ANSWER: ' h aa ss
PRACTICE: Complete 8 s a; to 12 s g.
This is the same as completing a; to 4 g, which can be computed by completing a; to g and adding 4.
ANSWER: 4 a;
PRACTICE: Estimate the area of a circle of diameter 27.
SOLUTION: First calculate l of 27.
Complete 3 to get 27.
The answer is 24.
Square 24.
576 is the area of the circle.
Redo with the square-first method.
ANSWER: 576, 569 s ds
PRACTICE: Find the area of a circle of radius 12. Then redo using the square-first method.
SOLUTION: First calculate l of 12.
Complete 1 d to get 12.
The answer is 10 '.
Square 10 '.
113 ' l is the area of the circle.
Redo with the square-first method.
ANSWER: 113 ' l, 112 s
PRACTICE: Find the area of a circle of radius 7. Then redo using the square-first method.
SOLUTION: First calculate l of 7.
Complete ' l to get 7.
This can be computed by completing ' l to 1 and adding 6.
The answer is 6 h ak.
Square 6 h ak.
Simplify dh gf dsf as parts of 324.
Redo with the square-first method.
ANSWER: 38 ' sj ka, 38 f ds
PRACTICE: Divide 15 by 42 in four lines.
ANSWER: d fs
ANSWER: ' ss hh
PRACTICE: Divide 11 by 21.
ANSWER: s fs
PRACTICE: Divide 31 by 2000.
ANSWER: a;; s;; s;;;
PRACTICE: Divide 66 by 840.
ANSWER: ag kf
PRACTICE: Divide 14 by 11 using completions.
ANSWER: 1 f ff
PRACTICE: Divide 8 h by 23 using completions.
ANSWER: d fh
PRACTICE: Finish the following multiplication by forming a completion and finishing in parts. Divide 8 g by 2 d.
SOLUTION:
ANSWER: 7 j;
PRACTICE: Consider a truncated pyramid with a 9-by-9 lower base, a 4-by-4 upper base, and a height of 8. What’s the volume?
Solution: Take the 4 and square it to get 16. Take the 9 and square it to get 81. Multiply the 4 and 9 to get 36.
The squares and products of the base lengths.
Add the 16, 81, and 36 to get 133. A third of the height is 2 '. Multiply 2 ' by 133 to get 354 ', the volume.
A third the height and the answer times the sum.
ANSWER: 354 '
PRACTICE: Given that 7 + 49 + 343 + 2,401 + 16,807 = 19,607, use the Egyptian trick to find 7 + … + 16,807 + 117,649.
SOLUTION: 7 × (19,607 + 1) = 7 × 19,608 = 137,256
ANSWER: 137,256
PRACTICE: A number and its s is added and it is 16. What is the number?
SOLUTION: Assume the number is 2.
Calculate 2 and its s to get 3.
2 and its :2 is 3.
Scale 3 to 16 by calculating 16 ÷ 3, which is 5 d.
16 ÷ 3 gives the scaling factor.
Scale up the assumed answer of 2 by a factor of 5 d to get the answer 10 '.
Scale up the false answer by 5 d.
ANSWER: 10 '
PRACTICE: A ninth is removed from a number and 20 remains. What is the number?
SOLUTION: Assume the number is 9, its ninth is 1 and when removed leaves 8.
Scale 8 to 20 by calculating 20 ÷ 8, which is 5 d.
20 × 8 gives the scaling factor.
Scale up the assumed answer of 9 by a factor of 2 s to get the answer 22 s.
Scale up the false answer by 2 s.
ANSWER: 22 s
PRACTICE: Five people get 35 loaves of bread, and each person gets 3 more than the person before. How much does each person get?
SOLUTION: The middle man gets the average of 35 ÷ 5.
The middle man gets 7. The two above each get 3 more, 10 and 13. The two below each get 3 less, 4 and 1.
ANSWER: 1, 4, 7, 10, and 13
PRACTICE: There are six people who get 33 loaves of bread. If each one gets 1 s more than the one before, how much does each person get?
SOLUTION: The average is 33 ÷ 6, which is 5 s.
The middle is 2 s jumps of 1 s from the start.
Since 2 s × 1 s is 3 s f, the first person gets 5 s - 3 s f. Complete f to 1 and add 1, which is 1 s f.
Adding 1 s to get the rest, the wages are 1 s f, 3 f, 4 s f, 6 f, 7 s f, and 9 f.
ANSWER: 1 s f, 3 f, 4 s f, 6 f, 7 s f, and 9 f.
REDO PRACTICE: Ninety loaves of bread get distributed between five people in arithmetic progression. If the upper three get 4 times as much as the lower two, how much does each person get?
SOLUTION: Assume they get a total of 60 loaves. The middle person gets 60 ÷ 5, or 12 loaves.
The lower two get one part while the upper three get four, so the lower two get a fifth of 60. So the lower two get a total of 60 ÷ 5, which is we already know is 12. The “middle person” of the two gets 12 ÷ 2, which is 6. The “middle person” of the first two makes 6 less than the middle man of all five and is 1 s jumps away, and hence each jump is 6 ÷ 1 s, or 4.
Adding and subtracting 4 from the middle person, we get 4, 8, 12, 16, and 20 loaves. We need to scale up our answer to 90 loaves of bread by multiplying by 90 ÷ 60, which is 1 s.
Now we multiply 1 s by each answer to get the adjusted shares.
Hence they get 6, 12, 18, 24, and 30.
ANSWER: 6, 12, 18, 24, and 30
PRACTICE: Sixty-six loaves of bread get distributed between five people in arithmetic progression. If the upper two get two times as much as the lower three, how much does each person get?
SOLUTION: Assume they get a total of 60 loaves. The middle person gets 60 ÷ 5, or 12 loaves.
The lower three get one part while the upper two get two, so the lower two get a third of 60. So the lower two get a total of 60 ÷ 3, which is 20.
The middle person of the three gets 20 ÷ 3, which is 6 '. The middle person of the first three makes 5 d less than the middle person of all five and is 1 jump away, and hence each jump is 5 d ÷ 1, or 5 d.
Adding and subtracting 5 d from the middle person, we get 1 d, 6 ', 12, 17 d, and 22 ' loaves. We need to scale up our answer to 66 loaves of bread by multiplying by 66 ÷ 60, which is 1 a;.
Now we multiply 1 a; by each answer to get the adjusted shares. (Add sums as parts of 30.)
Hence they get 1 d a; d;, 7 d, 13 g, 19 ag, and 24 ' g ag.
ANSWER: 1 d a; d;, 7 d, 13 g, 19 ag, and 24 ' g ag
PRACTICE: Represent the fl node of an odd-fraction family tree.
SOLUTION: 2 × fl = sk alh, 2 × sk = af, 2 × af = j, 2 × alh = lk, 2 × lk = fl
ANSWER:
PRACTICE: Find the complete odd-fraction family tree for sa.
SOLUTION: 2 × sa = af fs, 2 × af = j, 2 × fs = sa, 2 × j = f sk, 2 × f = s = *, 2 × sk = af
ANSWER:
PRACTICE: Use the previous answer to find the complete odd-fraction family tree for hd.
SOLUTION: 2 × hd = fs ash, above we know 42 goes to the sa tree. 2 × ash = hd.
ANSWER:
PRACTICE: Draw the augmented complete family tree of jg and show that it contains the fractions on the left-hand side of sg g; ag; = ag.
SOLUTION: 2 × jg = g; ag;, 2 × g; = sg, 2 × ag; = jg, 2 × sg = ag jg, 2 × ag = a; d;, 2 × a; = g, 2 × d; = ag, 2 × g = d ag
ANSWER:
PRACTICE: Express 10710 in base five and base-five calculi.
SOLUTIONS:
ANSWER: 4125
PRACTICE: Convert 3215 into base ten.
SOLUTION: Three quarters, 2 nickels, and 1 penny is worth (3 × 25) + (2 × 5) + 1 = 8610
ANSWER: 8610
PRACTICE: Using penny, nickel, and quarter calculi, add 2145 + 1235.
SOLUTION:
ANSWER: 3425
PRACTICE: Convert the following tokens into base four and base ten.
SOLUTION: In base four: 21314.
In base ten:
(2 × 64) + (1 × 16) + (3 × 4) + 1 = 128 + 16 + 12 + 1 = 15710
ANSWERS: 21314 and 15710
PRACTICE: Using base-seven calculi, add 3517 + 1327.
SOLUTION:
ANSWER: 5137
PRACTICE: Using base-three calculi, add 1203 + 2113.
SOLUTION:
ANSWER: 11013
PRACTICE: Add CXXVIII to LXXXXVIIII.
SOLUTION:
ANSWER: CCXXVII
PRACTICE: Add B%B%%V??C?MZ@ + B%%B%V??VCM#Z
ANSWER: NBVC??M##Z@@
PRACTICE: Convert the Sumerian calculi NB%%B%V?C??Z@ into Babylonian cuneiform.
SOLUTION: NB%%B% = 15 = Qt
V?C??= 23 = We and Z@ = 2 = w,
so the answer is Qt We w.
ANSWER: Qt We w
PRACTICE: Convert Qw Re Wo into base ten.
SOLUTION: The digits are 12, 43, and 29 so it’s equal to (12 × 602) + (43 × 60) + 29 = 43,200 + 2,580 + 29 = 45,80910.
ANSWER: 45,80910
PRACTICE: Translate 31⅔ into Babylonian.
SOLUTION: 31⅔ = 3140/60 = Eq.R
ANSWER: Eq.R
PRACTICE: How many degrees is q.Qw watches?
SOLUTION: q.Qw × 60 = 112/60 × 60
= (1 × 60) + (12/60 × 60)
= 60 + 12
= 72
ANSWER: 72 degrees
PRACTICE: Multiply Tr by Ru.
ANSWER: Rw Qi
PRACTICE: Find the length of the diagonal of a square whose side is Qu. Use common sense to place the decimal point.
ANSWER: Wr.w Wo R
PRACTICE: Multiply We by Qr Tt.
ANSWER: t Re t
PRACTICE: Multiply Tw Qo by w Eq Rt.
ANSWER: w Qw Qo e Qt
PRACTICE: Estimate the square root of 102.
SOLUTION: 102 = 100 + 2 = 102 + 2
ANSWER: 10
PRACTICE: Divide Re by Qw knowing that the reciprocal of Qw is t.
ANSWER: e.Et
PRACTICE: Use the above table to divide Q by Qy.
The Q row is Eu E. Since 10 ÷ 16 is just under 1, the decimal point goes in front.
ANSWER: .Eu E
PRACTICE: What is 10001102 in base ten?
SOLUTION: (1 × 64) + (0 × 32) + (0 × 16) + (0 × 8) + (1 × 4) + (1 × 2) + (0 × 1) = 7010
PRACTICE: Multiply 11102 by 1012 using binary calculi.
ANSWER: 10001102
PRACTICE: Use a one-column Egyptian style table to convert 55 into binary.
ANSWER: 1101112
PRACTICE: Convert d f g hekat into Horus Eye fractions.
SOLUTION: First convert into ro by multiplying by 320.
:3 :4 :5 hekat is 250 ' ro.
Then convert back to hekat by dividing by 320.
250 ' ro is :2 :4 :3:2 hekat and ' ro.
Now write the hekat fractions as Horus Eye fractions.
• 
hekat and ' ro.
ANSWER: 250 ' hekat is hekat and ' ro.
