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You start with the number 1. You can create a new number by applying an operation on two existing numbers (can be the same). The operations are +, - and *. What is the fewest number of steps needed to reach the number 123456? Bonus question: can you find multiple solutions?

Here is a similar puzzle: Creating 2020 in the fewest number of steps

Good luck!

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9 Answers 9

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10
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I have found some 9 step solutions

1.

1 + 1 = 2
2 + 1 = 3
3 * 2 = 6
6 * 3 = 18
18 + 1 = 19
19 * 18 = 342
342 + 19 = 361
361 * 342 = 123462
123462 - 6 = 123456

2.

1 + 1 = 2
2 + 1 = 3
3 * 2 = 6
6 * 3 = 18
18 + 1 = 19
19 * 18 = 342
342 * 19 = 6498
6498 * 19 = 123462
123462 - 6 = 123456

3.

1 + 1 = 2
2 + 1 = 3
3 * 2 = 6
6 * 3 = 18
18 + 1 = 19
19 * 19 = 361
361 * 19 = 6859
6859 * 18 = 123462
123462 - 6 = 123456

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5
  • $\begingroup$ Great work! This is the solution I had in mind. I wonder if it can be improved? $\endgroup$
    – Dmitry Kamenetsky
    Commented Dec 19, 2019 at 8:09
  • $\begingroup$ I am actually trying to simulate it on a computer. lets see If I find a better one. $\endgroup$
    – sudhackar
    Commented Dec 19, 2019 at 8:10
  • $\begingroup$ 9 seems to be the lower bound for what I see from the simulation $\endgroup$
    – sudhackar
    Commented Dec 19, 2019 at 8:11
  • $\begingroup$ I made a program to find all solutions and it didn't found the 8 step solution so nine step is minimum and 10 solutions exists $\endgroup$
    – Jirka Picek
    Commented Dec 19, 2019 at 9:25
  • $\begingroup$ Yea. The closest I got was to get 6*(19*19*19*3-1) to represent in 9 steps. $\endgroup$
    – sudhackar
    Commented Dec 19, 2019 at 9:42
8
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Minimum of steps is

9

I found 10 solutions:

Solution 1

1 + 1 = 2
1 + 2 = 3
3 * 2 = 6
6 * 3 = 18
18 + 1 = 19
19 * 18 = 342
342 + 19 = 361
361 * 342 = 123462
123462 - 6 = 123456

Solution 2

1 + 1 = 2
2 + 1 = 3
3 * 2 = 6
6 * 3 = 18
18 + 1 = 19
19 * 18 = 342
342 * 19 = 6498
6498 * 19 = 123462
123462 - 6 = 123456

Solution 3

1 + 1 = 2
2 + 1 = 3
3 * 2 = 6
6 * 3 = 18
18 + 1 = 19
19 * 19 = 361
361 * 18 = 6498
6498 * 19 = 123462
123462 - 6 = 123456

Solution 4

1 + 1 = 2
2 + 1 = 3
3 * 2 = 6
6 * 3 = 18
18 + 1 = 19
19 * 19 = 361
361 - 19 = 342
342 * 361 = 123462
123462 - 6 = 123456

Solution 5

1 + 1 = 2
2 + 1 = 3
3 * 2 = 6
6 * 3 = 18
18 + 1 = 19
19 * 19 = 361
361 * 19 = 6859
6859 * 18 = 123462
123462 - 6 = 123456

Solution 6

1 + 1 = 2
2 + 1 = 3
3 + 3 = 6
6 * 3 = 18
18 + 1 = 19
19 * 18 = 342
342 + 19 = 361
361 * 342 = 123462
123462 - 6 = 123456

Solution 7

1 + 1 = 2
2 + 1 = 3
3 + 3 = 6
6 * 3 = 18
18 + 1 = 19
19 * 18 = 342
342 * 19 = 6498
6498 * 19 = 123462
123462 - 6 = 123456

Solution 8

1 + 1 = 2
2 + 1 = 3
3 + 3 = 6
6 * 3 = 18
18 + 1 = 19
19 * 19 = 361
361 * 18 = 6498
6498 * 19 = 123462
123462 - 6 = 123456

Solution 9

1 + 1 = 2
2 + 1 = 3
3 + 3 = 6
6 * 3 = 18
18 + 1 = 19
19 * 19 = 361
361 - 19 = 342
342 * 361 = 123462
123462 - 6 = 123456

Solution 10

1 + 1 = 2
2 + 1 = 3
3 + 3 = 6
6 * 3 = 18
18 + 1 = 19
19 * 19 = 361
361 * 19 = 6859
6859 * 18 = 123462
123462 - 6 = 123456

Program which I made in C#:

using System;
using System.Collections.Generic;
using System.Linq;

namespace LowestPossible
{
    class Program
    {
        const int NUMBER_TO_FIND = 123456;
        const int MAX_DEPTH = 10;

        public static void Main()
        {
            var list = new List<Tuple<int, string>>();
            list.Add(new Tuple<int, string>(1, ""));
            rec(list, 1);
        }

        public static void rec(List<Tuple<int, string>> row, int step)
        {
            var lastRes = row.LastOrDefault();
            if (lastRes.Item1 == NUMBER_TO_FIND)
            {
                Console.WriteLine(String.Join("", row.Select(a => a.Item2 + " = " + a.Item1.ToString())) + " <" + (step-1) + ">");
                return;
            }

            if (step == MAX_DEPTH || lastRes.Item1 < 1)
            {
                return;
            }

            foreach (var num in row)
            {
                var newRow = new List<Tuple<int, string>>(row);
                newRow.Add(new Tuple<int, string>(lastRes.Item1 + num.Item1, " + " + num.Item1));
                rec(newRow, step + 1);
                newRow.RemoveAt(step);
                newRow.Add(new Tuple<int, string>(lastRes.Item1 - num.Item1, " - " + num.Item1));
                rec(newRow, step + 1);
                newRow.RemoveAt(step);

                newRow.Add(new Tuple<int, string>(lastRes.Item1 * num.Item1, " * " + num.Item1));
                rec(newRow, step + 1);
                newRow.RemoveAt(step);
            }
        }
    }
}

The output does not have nice format but it was made only for me. Also code is optimized to run in dotnetfiddle, but originaly it was for 2020 question. It runs too long for this question so it doesn't work for this one in dotnetfiddle due to timeout.

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8
  • $\begingroup$ There are multiple duplicates here, you can minimize that to 5 imo - like 5 and 10. $\endgroup$
    – sudhackar
    Commented Dec 19, 2019 at 9:26
  • $\begingroup$ Also is this the script? puzzling.stackexchange.com/a/92104/44046 $\endgroup$
    – sudhackar
    Commented Dec 19, 2019 at 9:35
  • 1
    $\begingroup$ I agree that getting 6 by 3+3 or 3*2 is nearly the same but it should be different solution since another operation is used $\endgroup$
    – Jirka Picek
    Commented Dec 19, 2019 at 9:36
  • $\begingroup$ No it's not the script, I made my own yesterday for that question but I didn't have time to publish the result and when I wanted to do so today, it was already published... I'll publish it later today $\endgroup$
    – Jirka Picek
    Commented Dec 19, 2019 at 9:38
  • $\begingroup$ Made a pretty substantial edit suggestion, feel free to reject and take parts if you like it $\endgroup$
    – Cireo
    Commented Dec 20, 2019 at 7:37
7
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I've found a 10-step solution:

 1 + 1 = 2
 2 + 1 = 3
 3 + 1 = 4
 4 + 4 = 8
 8 * 8 = 64
 8 + 2 = 10
 64 * 10 = 640
 640 + 3 = 643
 643 * 3 = 1929
 1929 * 64 = 123456

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7
  • $\begingroup$ Bravo. I figured out the 1929 trick, but couldn't make 1929 in few steps. $\endgroup$
    – Victor Stafusa
    Commented Dec 18, 2019 at 23:06
  • $\begingroup$ Math is hard, I agree. Seeing as how I sometimes calculate 0 times 3 = 3. $\endgroup$
    – Avi
    Commented Dec 18, 2019 at 23:15
  • $\begingroup$ Waiting for sw solution to prove it is optimal solution or find better one. $\endgroup$
    – z100
    Commented Dec 19, 2019 at 0:02
  • $\begingroup$ Very nice work! $\endgroup$
    – Dmitry Kamenetsky
    Commented Dec 19, 2019 at 0:06
  • 1
    $\begingroup$ @gustavovelascoh You can only use numbers you've already generated - the result of step 2 is useful later on, in steps 8 and 9. $\endgroup$
    – Avi
    Commented Dec 19, 2019 at 9:23
5
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I believe I can do it in 11 steps


1+1=2
2+2=4
4x4=16
16+16=32
16-1=15
16x16=256
256*15=3840
256*256=65536
65536-3840=61696
61696+32=61728
61728+61728=123456

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5
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On reflection I'm not sure this is the least steps, it's getting there though.

123455 steps

1+1=2
2+1=3
4=3+1
1+4=5
6=1+5
6+1=7
8=7+1
8+1=9
1+9=10
10+1=11
1+11=12
...
1+2023=2024
2025=1+2024
2025+1=2026
...
123452=1+123451
1+123452=123453
123454=123453+1
123455=1+123454
123454+1=123456

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2
  • 5
    $\begingroup$ You can save a step by using "+2" on the second to last step. $\endgroup$
    – Thomas Markov
    Commented Dec 19, 2019 at 17:49
  • $\begingroup$ You can add a step by using "+2" on the last step and then "-1". $\endgroup$
    – Stack Exchange Supports Israel
    Commented Dec 20, 2019 at 16:38
4
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Let's get the ball rolling. I doubt this is the fewest possible steps, but it should give people a baseline to try to beat.

Total steps: 12

$$ 1 + 1 = 2\\ 2 + 2 = 4\\ 4 + 4 = 8\\ 8 \times 8 = 64\\ 64 \times 8 = 512\\ 64 \times 64 = 4096\\ 4096 \times 2 = 8192\\ 4096 \times 8 = 32768\\ 32768 \times 4 = 131072\\ 131072 - 8192 = 122880\\ 122880 + 512 = 123392\\ 123392 + 64 = 123456\\ $$

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2
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Two different 10-step solutions I found (without programming) are:

1 + 1 = 2
1 + 2 = 3
2 + 2 = 4
4 + 4 = 8
2 + 8 = 10
8 * 8 = 64
3 * 64 = 192
64 * 10 = 640
640 + 3 = 643
192 * 643 = 123456

and

1 + 1 = 2
2 + 2 = 4
4 + 2 = 6
4 * 4 = 16
16 * 16 = 256
256 - 1 = 255

16 * 6 = 96
255 + 96 = 351

351 * 351 = 123201
123201 + 255 = 123456

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2
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A 9 step solution that doesn't use subtraction:

1+1 = 2
2+2 = 4
4*4 = 16
16+4 = 20
20+4 = 24
16*16 = 256 or 16*20 = 320
256*20 = 5120 or 320*16 = 5120
5120+24 = 5144
5144*24 = 123456

Another 9 step solution nobody has mentioned:

1+1 = 2
2+2 = 4
2+4 = 6
4*4 = 16
16*6 = 96
96-16 = 80 or 16-96 = -80
16*80 = 1280 or 16*-80 = -1280
1280+6 = 1286 or 6 - (-1280) = 1286
1286*96 = 123456

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4
  • $\begingroup$ Beautiful and original! After seeing this I removed the tick. So it's back up for grabs. $\endgroup$
    – Dmitry Kamenetsky
    Commented Dec 21, 2019 at 8:00
  • $\begingroup$ My search found only these and the ones Jirka/sudhacker found (plus minor variants on those, such as swapping the order of 6/16). However, I constrained my search to numbers between -140000 and 140000, so there may be additional answers that go outside those bounds. $\endgroup$
    – ralphmerridew
    Commented Dec 22, 2019 at 12:26
  • $\begingroup$ My random search got these too, but somehow I did not minimize. $\endgroup$
    – sudhackar
    Commented Dec 24, 2019 at 7:46
  • $\begingroup$ My code: function rseek (ar) { if (ar[ar.length-1] == 123456) { console.log(ar+""); return; } if (ar.length == 10) return; var successes = get_nexts(ar); for (var a in successes) { if (-140000 <= a && a <= 140000) { ar.push (a-0); rseek(ar); ar.pop(); } } } function get_nexts(list) { var rv = {}; for (var a of list) for (var b of list) rv[a+b] = rv[a*b] = rv[a-b] = 1; for (var a of list) delete rv[a]; return rv; } $\endgroup$
    – ralphmerridew
    Commented Jan 10, 2020 at 21:57
1
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$6*6=36$
$36*4=144$
$144*144=20736$
$20736*6=124416$
$5*6=30$
$30*30=900$
$30*2=60$
$900+60=960$
$124416-960=123456$

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