"I don't know.", said the cook, "but every 2 used a dish of rice, every 3 used a dish of broth, and every 4 used a dish of meat between them". There were 65 dishes in all. How many guests were there?
Let us investigate some earlier ones.
The first three guests used a dish of rice and a dish of broth; that would be two dishes.
The fourth used used a dish of rice and a dish of meat; that would be two more dishes.
The sixth used a dish of rice, and a dish of broth; that would be two more dishes.
The eight used a dish of rice, and a dish of meat; that would be two more dishes.
The ninth used a dish of broth; that would be one more dish.
The tenth used a dish of rice; that would be one more dish.
The twelfth used a dish of rice, a dish of broth, and a dish of meat; that would be three more dishes.
I think we can in fact find an iterative summation algorithm here:
- dishCount = 0
- numGuests = 1
- while (dishCount < 65):
- if (numGuests is a multiple of 2):
- dishCount += 1
- Immediately exit loop if dishCount is at or exceeds 65
- if (numGuests is a multiple of 3):
- dishCount += 1
- Immediately exit loop if dishCount is at or exceeds 65
- if (numGuests is a multiple of 4):
- dishCount += 1
- Immediately exit loop if dishCount is at or exceeds 65
- print("The number of guests is: " + str(numGuests))
Asking if a given integer is a multiple of n is equivalent to asking if the said integer does not produce a remainder when divided by n.
Thus, we arrive at the following Python code:
dishCount = 0
numGuests = 1
while (dishCount < 65):
print("Current number of guests: " + str(numGuests) + ", current dishes served: " + str(dishCount))
if (numGuests%2 == 0):
dishCount += 1
print("Current number of guests: " + str(numGuests) + ", dish of rice served.")
if (dishCount >= 65):
continue
if (numGuests%3 == 0):
dishCount += 1
print("Current number of guests: " + str(numGuests) + ", dish of broth served.")
if (dishCount >= 65):
continue
if (numGuests%4 == 0):
dishCount += 1
print("Current number of guests: " + str(numGuests) + ", dish of meat served.")
if (dishCount >= 65):
continue
numGuests += 1
print("The number of guests is: " + str(numGuests))
The program prints 60:
Current number of guests: 1, current dishes served: 0
Current number of guests: 2, current dishes served: 0
Current number of guests: 2, dish of rice served.
Current number of guests: 3, current dishes served: 1
Current number of guests: 3, dish of broth served.
Current number of guests: 4, current dishes served: 2
Current number of guests: 4, dish of rice served.
Current number of guests: 4, dish of meat served.
Current number of guests: 5, current dishes served: 4
Current number of guests: 6, current dishes served: 4
Current number of guests: 6, dish of rice served.
Current number of guests: 6, dish of broth served.
Current number of guests: 7, current dishes served: 6
Current number of guests: 8, current dishes served: 6
Current number of guests: 8, dish of rice served.
Current number of guests: 8, dish of meat served.
Current number of guests: 9, current dishes served: 8
Current number of guests: 9, dish of broth served.
Current number of guests: 10, current dishes served: 9
Current number of guests: 10, dish of rice served.
Current number of guests: 11, current dishes served: 10
Current number of guests: 12, current dishes served: 10
Current number of guests: 12, dish of rice served.
Current number of guests: 12, dish of broth served.
Current number of guests: 12, dish of meat served.
Current number of guests: 13, current dishes served: 13
Current number of guests: 14, current dishes served: 13
Current number of guests: 14, dish of rice served.
Current number of guests: 15, current dishes served: 14
Current number of guests: 15, dish of broth served.
Current number of guests: 16, current dishes served: 15
Current number of guests: 16, dish of rice served.
Current number of guests: 16, dish of meat served.
Current number of guests: 17, current dishes served: 17
Current number of guests: 18, current dishes served: 17
Current number of guests: 18, dish of rice served.
Current number of guests: 18, dish of broth served.
Current number of guests: 19, current dishes served: 19
Current number of guests: 20, current dishes served: 19
Current number of guests: 20, dish of rice served.
Current number of guests: 20, dish of meat served.
Current number of guests: 21, current dishes served: 21
Current number of guests: 21, dish of broth served.
Current number of guests: 22, current dishes served: 22
Current number of guests: 22, dish of rice served.
Current number of guests: 23, current dishes served: 23
Current number of guests: 24, current dishes served: 23
Current number of guests: 24, dish of rice served.
Current number of guests: 24, dish of broth served.
Current number of guests: 24, dish of meat served.
Current number of guests: 25, current dishes served: 26
Current number of guests: 26, current dishes served: 26
Current number of guests: 26, dish of rice served.
Current number of guests: 27, current dishes served: 27
Current number of guests: 27, dish of broth served.
Current number of guests: 28, current dishes served: 28
Current number of guests: 28, dish of rice served.
Current number of guests: 28, dish of meat served.
Current number of guests: 29, current dishes served: 30
Current number of guests: 30, current dishes served: 30
Current number of guests: 30, dish of rice served.
Current number of guests: 30, dish of broth served.
Current number of guests: 31, current dishes served: 32
Current number of guests: 32, current dishes served: 32
Current number of guests: 32, dish of rice served.
Current number of guests: 32, dish of meat served.
Current number of guests: 33, current dishes served: 34
Current number of guests: 33, dish of broth served.
Current number of guests: 34, dish of rice served.
Current number of guests: 35, current dishes served: 36
Current number of guests: 36, current dishes served: 36
Current number of guests: 36, dish of rice served.
Current number of guests: 36, dish of broth served.
Current number of guests: 36, dish of meat served.
Current number of guests: 37, current dishes served: 39
Current number of guests: 38, current dishes served: 39
Current number of guests: 38, dish of rice served.
Current number of guests: 39, current dishes served: 40
Current number of guests: 39, dish of broth served.
Current number of guests: 40, current dishes served: 41
Current number of guests: 40, dish of rice served.
Current number of guests: 40, dish of meat served.
Current number of guests: 41, current dishes served: 43
Current number of guests: 42, current dishes served: 43
Current number of guests: 42, dish of rice served.
Current number of guests: 42, dish of broth served.
Current number of guests: 43, current dishes served: 45
Current number of guests: 44, current dishes served: 45
Current number of guests: 44, dish of rice served.
Current number of guests: 44, dish of meat served.
Current number of guests: 45, current dishes served: 47
Current number of guests: 45, dish of broth served.
Current number of guests: 46, current dishes served: 48
Current number of guests: 46, dish of rice served.
Current number of guests: 47, current dishes served: 49
Current number of guests: 48, current dishes served: 49
Current number of guests: 48, dish of rice served.
Current number of guests: 48, dish of broth served.
Current number of guests: 48, dish of meat served.
Current number of guests: 49, current dishes served: 52
Current number of guests: 50, current dishes served: 52
Current number of guests: 50, dish of rice served.
Current number of guests: 51, current dishes served: 53
Current number of guests: 51, dish of broth served.
Current number of guests: 52, current dishes served: 54
Current number of guests: 52, dish of rice served.
Current number of guests: 52, dish of meat served.
Current number of guests: 53, current dishes served: 56
Current number of guests: 54, current dishes served: 56
Current number of guests: 54, dish of rice served.
Current number of guests: 54, dish of broth served.
Current number of guests: 55, current dishes served: 58
Current number of guests: 56, current dishes served: 58
Current number of guests: 56, dish of rice served.
Current number of guests: 56, dish of meat served.
Current number of guests: 57, current dishes served: 60
Current number of guests: 57, dish of broth served.
Current number of guests: 58, current dishes served: 61
Current number of guests: 58, dish of rice served.
Current number of guests: 59, current dishes served: 62
Current number of guests: 60, current dishes served: 62
Current number of guests: 60, dish of rice served.
Current number of guests: 60, dish of broth served.
Current number of guests: 60, dish of meat served.
The number of guests is: 60
I think I can whip out a formula at this point:
d/2 + d/3 + d/4 = 65
6d + 4d + 3d = 780
13d = 780
d = 60
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