# Rendering the number 10,958 with the string 1 2 3 4 5 6 7 8 9

Brazilian mathematician Inder Taneja has found a way of expressing every number between 1 and 11,111, except 10,958, by inserting mathematical operators in between the numbers 1 2 3 4 5 6 7 8 9 and evaluating the expression. He did so using the four basic arithmetic operations, exponentiation, concatenation, and brackets, but avoiding factorials, square roots, and decimals. If these last three operations are allowed, can 10,958 be likewise expressed?

• Do the digits 1 2 3 4 5 6 7 8 9 have to be in order? Jan 12, 2017 at 22:10
• Yes, they have to be in order. Jan 12, 2017 at 22:11
• Link to Taneja's paper for those who are interested in all the other numbers: arxiv.org/pdf/1302.1479v5.pdf Jan 12, 2017 at 22:30
• This is a "decreasing" solution: 9 8 7.... 1 rather than increasing. Jan 13, 2017 at 12:48
• @Bernardi Yes, I deleted my comment when I saw it Jan 13, 2017 at 12:51

Taking from

$(1 + 2 + 34) \times (5 × 6 + 7) \times 8 + 9 = 10961$

We have

$(1 + 2 + 34) \times (5 × 6 + 7) \times 8 + (\sqrt{9})! = 10958$

• Ha! I hadn't even thought of looking at the other solutions and finding one I could tweak! Here I was racking my brain from scratch! :) Nice one! +1 Jan 13, 2017 at 7:06
• +1 for the attempt. But don't you need to have an operator in between each number i.e you are using 34 directly? Jan 13, 2017 at 8:28
• Going by the OP, concatenation is allowed. Jan 13, 2017 at 9:09
• The solution uses both square roots and factorials, so it's not quite in the spirit of the paper.
– user36263
Apr 26, 2017 at 4:13

Voila! With arithmetic operators, factorial and exponents:

$\mathbf{{(1+2^3)}^4 + 5 - 6! + 7! + (8 \times 9 )} = {(1+8)}^4 + 5 - 720 + 5040 + 72 = 6561 + 4397 = 10958$

• A great answer using no square roots and concatenation. Good Job. Jan 13, 2017 at 10:14
• The brackets around $(8 \times 9)$ are totally unecessary. Dec 21, 2017 at 4:49

This is as close as I can get:

(1+2)3+4 * 5 + (6 * 7) - 8 - 9 = 10960.

Maybe this will give someone an idea of how to get there!

Concatenation is ||

1 x 23 + ((4x5x6)||7 + 8) x 9

• This checks out. QUESTION: Am I misreading the OP or does this solution satisfy the Taneja requirements without adding extra operators? since concatenation was allowed to begin with. Oct 21, 2017 at 13:36

Matt Parker from the Numberphile YouTube channel found a solution and explains it in this video.

|| stands for concatenation. (ie: 1||0 x 2 = 20)

1 x 23 + ((4x5x6)||7 + 8) x 9

Concatenation is heavily used in the paper but I don't think it ever have been used that way. (ie:(2 x 3) || 2 = 62)

Does it's stand with the spirit of the paper? It's debatable.

The closest I can get

$$(12*3*4÷5*6*7+8)*9=10958.4$$