# Enigmatic Code

Programming Enigma Puzzles

## Enigma 1358: Five fives

From New Scientist #2517, 17th September 2005

You might think there is something wrong with the addition sum shown below, but in fact each of the five numbers shown in the sum is in a different base. All five numbers are even and the given total at the bottom is less than 100,000.

What is that total?

[enigma1358]

### 3 responses to “Enigma 1358: Five fives”

1. Jim Randell 4 January 2014 at 9:08 am

This is a relatively straightforward puzzle, but I think it would act as a good introductory puzzle for someone just starting to do Enigmas.

This Python program runs in 33ms.

```from itertools import count, combinations
from enigma import concat, printf

# find 11101 in various bases
# (we only need to consider odd bases as the numbers are even)
ns = list()
for b in count(3, step=2):
n = int('11101', base=b)
if not(n < 100000): break
ns.append(n)
printf("[b={b} n={n}]")

# find 4 of the numbers that sum to a fifth
for s in combinations(ns[:-1], 4):
n = sum(s)
if n in ns:
printf("{s} = {n}", s=concat(*s, sep=' + '))
```

Solution: The total of the sum is 88724 (in base 10, in the puzzle it is written in base 17).

• Naim Uygun 4 January 2014 at 12:07 pm
```# the bases are 5,7,13,15,17
# 776 + 2794 + 30928 + 54226 = 88724
from itertools import combinations
for a,b,c,d,e in combinations(range(2,20),5):
if max(a,b,c,d,e)==e :
n5= int('11101', base=e)
if  n5 < 100000 :
#print(a,b,c,d,e)
n1=int('11101', base=a)
n2=int('11101', base=b)
n3=int('11101', base=c)
n4=int('11101', base=d)
t=n1+n2+n3+n4
if t==n5:
print("The bases are ",a,b,c,d,e)
print(n1,"+",n2,"+",n3,"+",n4,"=",n5)
break
input("THE END")
```
• Brian Gladman 7 January 2014 at 11:11 pm

Hi Naim,

I thought that you might be interested to know that your line

```if max(a, b, c, d, e) == e:
```

is not neccsary because combinations are returned in (lexicographic) order whenever the input is in this order. In your use this means that the test is not necessary since e will always be the largest of a, b, c, d and e.

Brian

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