Le Monde puzzle [#814]

The #814 Le Monde math puzzle was to find 100 digits (between 1 and 10) such that their sum is equal to their product. Given the ten possible values of those digits, this is equivalent to finding integers a₁,…,a₁₀ such that

a₁+…+a₁₀=100

and

a₁+2a₂+…+10a₁₀=2^a₂x….x10^a₁₀,

which reduces the number of unknowns from 100 to 10 (or even 9). Furthermore, the fact that the (first) sum of the a_i‘s is less than 100 implies that the (second) sum of the ia_i‘s is less than 1000, hence i^a_i is less than 1000. This reduces the number of possible ten-uplets enough to allow for an enumeration,  hence the following R code:

bounds=c(100,trunc(log(1000)/log(2:10)))

for (i2 in 0:bounds[2])
for (i3 in 0:bounds[3])
for (i4 in 0:bounds[4])
for (i5 in 0:bounds[5])
for (i6 in 0:bounds[6])
for (i7 in 0:bounds[7])
for (i8 in 0:bounds[8])
for (i9 in 0:bounds[9])
for (i10 in 0:bounds[10]){

  A=c(i2,i3,i4,i5,i6,i7,i8,i9,i10)
  if (sum(A)<101){

   A=c(100-sum(A),A)
   if (sum((1:10)*A)==prod((1:10)^A))
     print(A)
  }}

that produces two answers

 [1] 97  0  0  2  0  0  1  0  0  0
 [1] 95  2  3  0  0  0  0  0  0  0

i.e. either 97 1’s, 2 4’s and 1 7, or 95 1’s, 2 2’s and 3 3’s. I would actually love to see a coding solution that does not involve this pedestrian pile of “for”. And a mathematical solution based on Diophantine equations. Rather than the equally pedestrian solution given by Le Monde this weekend.