# Sunday Times Teaser 2809 – This and That

*by Victor Bryant*

#### Published: 24 July 2016 (link)

I have written down a list of eleven numbers and then consistently replaced digits by letters, with different letters for different digits. In this way the list becomes:

**SO DO WHAT NOW ADD AND SEND IN THIS AND THAT**

The grand total of these eleven numbers is a four-figure number.

Numerically, what is **THIS** + **THAT**?

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A manual solution can be found by expressing the sum of the eleven numbers in terms of the letters used as: 2002.T + 1011.S + 1001.W + 320.A + 300.H + 131.N + 100.E + 24.D + 20.I + 12.O.

To find the minimum of this sum, we need to allocate letters in inverse proportion to the values by which they are multiplied so that small letter values go with large multipliers. But, from left to right, H is the first letter that is not a leading digit and can hence be zero. So the allocations we need are T = 1, S = 2, W = 3, A = 4, H = 0, N = 5, E = 6, D = 7, I = 8 and O = 9, giving a minimum sum of 9998.

Since any other letter/digit allocation will give a sum at least four higher with five digits, this minimum is the only possible four digit sum and must hence provide the solution. So THIS + THAT = 2123.

The general Alphametic solver that I recently added to the

enigma.pylibrary solves this problem in 1.4 seconds.This is sufficiently fast that I don’t feel the need to write a specific solver for this puzzle.

enigma.pyis available at [https://www.magwag.plus.com/jim/enigma.html].See [ https://enigmaticcode.wordpress.com/2016/06/29/solving-alphametics-with-python-part-2/ ] for a description of the Alphametic solver.

It’s also encouraged me to add a “–answer=” option to the solver, which saves you the bother of doing the final bit of arithmetic to get the answer to the problem:

Incidentally, the next smallest possible sum would be 10002.

This gives the result: THIS + THAT = 2123 (TOTAL = 9998, THIS = 1082, THAT = 1041, SO = 29, DO = 79, IN = 85, NOW = 593, ADD = 477, AND = 457, SEND = 2657, WHAT = 3041).